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ORNL~4674
UC~-80 — Reactor Technology

Contract No, W-7405-eng~26

REACTIVITY BALANCE CALCULATIONS AND LONG~TERM
REACTIVITY BEHAVIOR WITH 23°U TN THE MSRE

B. E. Prince
J. R. Engel
C. H. Gabbard

FEBRUARY 1972

0AK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee 37830
operated by
UNTON CARBIDE CORPORATION
for the
U.S. ATOMIC ENERGY COMMISSION

LGCKHMEED MART iy EN

DT

.’l i
3 4456 D515983 a

ERGY RESEARCH | t3RARE

i
ABSTRACT
1. INTRODUCTION

2. THEORETICAL FOUNDATIONS OF THE REACTIVITY BALANCE

2.1 General Considerations

2.2 Simplifications

i1i

CONTENTS

2.3 Approximate Techniques in Calculation .

2.4 Reactivity Measurements

3. DESCRIPTION OF INDIVIDUAL REACTIVITY EFFECTS IN THE MSRE

3.1 Reference Conditions

3.2 The General Reactivity Balance Equation .

3.3 Influence of Graphite Irradiation Damage on the MSRE

Reactivity Balance

4. SOURCES OF DATA AND PARAMETER CALCULATIONS
4.1 Initial Nuclide Concentrations

4.2 Average Reaction Cross Sections

4.3 Neutron Flux Normalization

4.4 Flux—Adjoint Flux Integrals and Reactivity Parameters

4.5 Effective Delayed-Neutron Fractions .
5. EXPERIENCE WITH THE REACTIVITY BALANCE

5.1 Brief History of Nuclear Overations with 23°U .

5.2 Results of On~Line Reactivity Balance Calculations

5.3 Long~Term Reactivity Trends

6. CONCLUSIONS
REFERENCES . . .
APPENDIX .

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ABSTRACT

Reactivity trends in the Molten-S5alt Reactor Experiment were studied
from the beginning of power operation by comparing predicted effects of
changes in the core conditions with observed compensating effects asso-
ciated with control rod motion. The calculations were made on~line, during
operation by using an available digital-computer data logger. The changes
accounted for included uranium depletions and additions; plutonium produc-
tion; 135Xe, samarium, and other fission product production; temperature
variations; and burnout of residual ®Li and !B (in graphite).

The entire period of operation with 2?°U in the fuel salt was studied,
and the essential conclusions were as follows.

1. 1In the normal range of reactor operating conditions, the magnitude
of residual (unaccounted for) reactivity always remained less than 0.1%
Sk/k; its rate of variation over the sampling intervals was small and
essentially random (*0.01%Z 8k/k) and gave no indication of instability of
the fuel composition.

2. The largest "abnormal' variations observed in the residual re-
activity were less than 0.2% Sk/k and appeared to be associated with
cnanges in xenon poisoning and the entrained circulating gas in the core
at scheduled off-normal system operating conditions. Although not ac-
counted for by the on-line calculations, these variations were in accord
with the qualitative behavior expected under these conditions and hence
were not regarded as evidence of malfunctions. At no time did the magni-
tude of residual reactivity approach the administrative limit established
at the start of operations (0.5% 8k/k, which is the approximate value of
the delayed~neutron fraction with the fuel circulating).

3. The apparent stability of other conditions in the core which
could potentially affect the reactivity allowed the reactivity balance
to be used as an effective diagnostic tool for studying the behavior
of !'3%Xe and effects of cover~gas entrainment in the fuel salt.

4. The long-term variation in residual reactivity during the entire
period of operation with ?°°U was less than 0.1% 8k/k. Analysis showed
that this amount of variation could be ascribed to the combined uncertain-
ties in thermal power level, fission product removal from the systen,

and slight structural changes in the graphite during irradiation.
1. INTRODUCTION

A feature of the nuclear operations analysis of the Molten Salt Re-
actor Experiment (MSRE) that is not yet commonly used was on-line reac-—
tivity balance calculations to serve the dual purpose of malfunction
detection and a means of determining the predictability of the long-term
nuclear performance of the reactor. MSRE nuclear operations were termi-
nated after about four years of successful power operation encompassing
two successive fuel loadings, one with 235y and the other with 2°%U. This
report describes experience with the reactivity balance calculation as a
tool of nuclear operations analysis.

The MSRE was built to demonstrate the feasibility and operability of
a high-temperature reactor with a mixture of molten fluoride salts as the
circulating fluid fuel. A major consideration in the safety of any fluid-
fueled reactor is the chemical and physical stability of the fuel mixture.
Because of the mobility of the fuel, a considerable effort was required
during all phases of the reactor program to assure proper fuel behavior
and to attend to the possibility, consequences, and detection of any sep-
aration of fissionable material from the mixture. Although there was no
known mechanism for fuel separation within a wide range of conditions
around the operating point, hypothetical consequences of uranium separa-
tion and redispersion were studied extensively in the early safety analy-
sis of the reactor. These studies formed the main original incentive for
setting up a program to monitor the nuclear reactivity of the operating
system for variations that might suggest abnormal behavior of the fuel.

An operating limitation of *0.5% 6k/k, was also placed on any reactivity
anomaly. As time went on and operating experience was accumulated, no
evidence of any instability of fuel behavior (physical or chemical) was
apparent, and interest tended to shift toward the use of the reactivity
balance calculation as an aid in analysis of normal nuclear operations.

The use of reactivity trends for monitoring and analyzing the state
of the reactor actually has had a long history in power reactor develop-
ment. Perhaps the classic example is the first detection and identifica-
tion of the reactivity poisoning by 13%%e in the Hanford reactors by Fermi,

Wheeler, and others, as discussed by Weinberg and Wigner.l Generally,
however, routine applications of this technique tended fo remain dormant
while refinements in the techniques of measurement and mathematical model-
ing of the nuclear performance were being developed. With the advent of
on-line digital computers as a means of fast data logging and processing,
interest in the potential of the reactivity balance calculation is increas-
ing. The recent article by Danforth? gives a good status summary of con-
temporary applications.

In this report of the use of reactivity balance calculations, the
first section consists of a general description of the theoretical founda~
tions of the technique. Study of the literature indicated that elements
of the mathematical description of this method existed largely in fragments
that emphasized other applications. Hence Section 2 was written with the
intent of clarifying the rules for using the reactivity concept in a con-
gsistent manner for this application. Section 2 is expository in character,
and readers who feel sufficiently familiar with the concept of reactivity
can omit this section without loss of continuity. Section 3 describes
reactivity effects in the MSRE and the procedures used for mathematical
modeling of them. The sources of data and calculated parameters that are
the required input in the reactivity balance model are discussed in Section
4. The final sections describe the results of applying this technique in
the MSRE nuclear operations analysis and include a discussion of the in-
terpretation of the reactivity balance data and a retrospective view of
the usefulness and limitations of the technique in application to molten-

salt reactor analysis.
2. THEORETICAL FOUNDATIONS OF THE REACTIVITY BALANCE

2.1 General Considerations

Fundamental principles require that for steady power operation of a
nuclear reactor a balance be maintained between the rates of neutron pro-
duction and disappearance due to absorptions and leakage to the surround-
ings. Stated loosely, reactivity is a conceptual quantity, introduced to
describe tramsient situations in which these rates do not balance. It is
convenient to express this quantity as the ratio of net to total production

rates of neutrons in the entire reactor; that is

total production rate — total rate

Reactivit -
Re Y total production rate

. (1)
Thus reactivity is a defined quantity that is susceptible to measurement
only by indirect means, and only in this indirect sense does it make "physi-
cal" appearance when the neutron population is changing. If the population
or power level is at steady state in the absence of any neutron source
other than the fission chain reaction, the reactivity must be zero, and
any attempt to ascribe separate reactivity components to the reactor state
at a given instant or to changes during an interval of steadv operation is
merely a convenient bookkeeping device. However, if procedures are de-
veloped for consistent use of this device to monitor reactor operation and
it is found that the algebraic sum of these component reactivities deviates
from zero, this can mean either that the calculations of individual known
effects are in error or that there are unknown or anomalous changes occur-
ring that are not accounted for inm the calculations. Power operation of
a reactor is a complex situation in which many effects simultaneously in-
fluence the neutron reaction rates. The device of separating these effects
according to a reactivity scale allows individual experiments or computa-
tions to be used as an aid in interpreting the whole process.

While reactivity is a central concept in nuclear operations analysis,
the fact that it is an inferred quantity can give rise to inconsistencies

in applications. 1In the reactivity balance, we are concerned with the
theory of compensating reactivity changes that are generally not small,

in contrast to the many routine applications of the reactivity concept in
which only small, uncompensated changes are considered. This requires a
precise examination of the sense in which reactivity effects are additive,

With the present state of the art, neither an approach based purely
on theoretical calculations nor one based entirely on experimental measure-
ments is practical. Instead, an operational reactivity balance must con~
sist of a mixture of these two types of evaluation. Thus the constructlion
of the reactivity balance also involves related problems of interpreting
individual reactivity measurements.

A specific example, which actually has practical application in inter-
preting the MSRE rod calibration experiments, can serve as a prototype in
discussing the theoretical construction of the reactivity balance. 1In
Fig. 1 the parameter region defining an Important part of the rod calibra-

tion measurements is shown schematically. Consideration of variations in

ORNL—DWG 71-9805

 

 

>._
=
=
'_
O
<L
wl
o
o CONTRQOL ROD POSITION
7t
S 7
7
7
/
b / |
/ |
|
/
| \g\ s a
| éfi /
| O /
| & ’
| /& /
& ~ CRITICALITY LINE
<(/+ ““““““““““““““ T S— c

Fig. 1. Parameter region in zero-power rod~calibration experiments.
core temperature, shim rod positions, xenon and samarium poisoning, and
other nuclide changes during power operation have been temporarily sup-
pressed, and we have shown only the effect of varying the position of the
regulating rod as the fuel salt is enriched with excess fissile uranium
(beyond the amount required for criticality with the rods fully withdrawn),
The vertical dimension in this "three-dimensional™ plot is a reactivity
scale. The locus of points describing the change in critical rod position
as excess fissile uranium is added to the reactor is shown as the solid
curve oc lying in the horizontal plane (reactivity = 0). This locus is
a graphical expression of the reactivity balance for this parameter re-
gion, We may further observe that this curve (rod position vs uranium
concentration) is the only relation directly accessible to us from experi-
ment provided we confine ourselves to critical states of the reactor.
Suppose, now,'that we wish to extend this description to the case where
other parameters mentioned above may also vary during reactor operation.
This would correspond to a path of critical states in a multidimensional
space analogous to Fig. 1. In this situation, changes in parameters other
than those represented along the axes of independent variables of Fig. 1
can balance nuclear effects of changes along these axes in order to produce
a critical state. This provides a basic motive for mapping the effects
of individual changes in rod position or uranium concentration on the re-
activity scale, even though only stationary states of the reactor flux are
to be described in this application.* This would be represented by knowl-
edge of the complete surface, 5, in Fig. 1. To obtain this information,
it is necessary to ''reach" into the vertical dimension, either by per-
fofming kinetic experiments or by performing calculations for the off~
eritical states.

The static reactivity concept is often used to describe the off-
critical state. We shall sketch here only the essential features of this

. + - -I- - »
theoretical construction as they relate to this problem. In this concept

 

%
Separate motives can be associated with the needs for quantitative

description of the kinetic behavior of the reactor under independent
parameter variations.

+A much more thorough description is given in Chap. XIT of ref. 1.
the set of physical critical states (the curve oc in Fig. 1) is embedded
in a more general set of pseudocritical states in which the actual neutron
production per fission, v, is multiplied by a calculated factor 1 ~ Pg,
determined to make the neutron production and loss rates balance. The
quantity p. is defined to be the static reactivity. TIf the basic theo-
retical model describing the neutron transport processes in the reactor

is sufficiently realistic for the critical state, use of the calculated
static reactivities for the off-critical states gives one possible opera-
tional description of the surface S in Fig. 1. Leaving aside, for con-~
ceptual purposes, the problems of determing what is a sufficiently real-
istic model for computations (e.g., diffusion theory vs transport theory,
etc,) and assuming that our model gives an exact description of the criti-
cal state, it is possible to relate the static reactivity for an off-
critical state in a precise way to the reactivity inferred from kinetic
measurements [via Eq. (1)]. This relation is nontrivial and has been the
subject of several fundamental papers in the reactor physics literature,*
For the purposes of this review, we will merely state that certain kinetic
experiments, such as period-differeatial worth and rod-drop integral-worth
experiments, can be designed and analyzed (introducing caleculated correction
factors, if necessary) so that differeunces in the reactivities are opera-
tionally insignificant. (Specific details relevant to the application of
MSRE measurements in the reactivity balance are described later in this
report.) Therefore, as a framework for mathematical description of the
theory of compensating reactivity effects, we can use the static reactivity
concept.

Two expressions derived for the static reactivity corresponding to an
arbitrary reactor stafte are given in the Appendix of this report, starting
with the basic equation describing the balance of neutron production and
loss processes. Here, use is made of linear operator notation in order to
retain the basic mathematical structure descriptive of the physical pro-
cesses, while avoiding cumbersome notation associated with choice of a par-
ticular description (e.g., the Boltzmann transport equation or the multi-

group diffusion equations). The first expression derived ianvolves only

 

References 3, 4, and 5 are judged classical in this category.
the quantities descriptive of production and loss processes for a single

state integrated over the reactor volume; namely,

(W,49) + (¥,L4)
p, = 1-— . (2)
(¥, P9)

 

In this general notation, L, 4, and P represent linear operators governing
the neutron leakage, absorption (including energy transfer by scattering),
and production from fission respectively. The functions ¢ and Yy are the
direct and adjoint flux fields corresponding to that reactor state, and
the symbol, for example, (y,Ld), represents the inner product of the func-
tions ¢ and Lé4. (The inner product of two functions is defined here as the
multiplication of the functions and integration over the spatial, energy,
and, in certain applications, the directional variables describing the
flux fields.)* Further explicit use of Eq. (2) will not be necessary in
our application; it is intrinsically contained in the relations for reac-
tivity, however, because it defines in an absolute sense the static reac-
tivity corresponding to some chosen reference state of the reactor,

The second expression is relative in character and involves quantities
descriptive of two states, which we will arbitrarily designate "initial"
and "final' states. 1If we use subscript zero to distinguish quantities

corresponding to the initial state, this formula may be written

(5,8P0) (b ,848) (¥ ,6L0)
P, "0, = (L—=p) - - : (3a)
(d)os‘- qu) (wO,PO(b) (waPO‘i’)

An equivalent expression is useful in those situations where the production
operator is altered and the reactivity in the final state, Ogs is con~

sidered to be an unknown quantity. This is,

 

In applying this notation to the formalism of transport theory, ¢
and ¢ must be thought of as angular fluxes, and the leakage term, L¢, can
be taken as R-V$, However, these observations are not central to our dis-
cussion, particularly if the multigroup diffusion model is to he applied,
where the angular variables have been integrated out.
(¢O,6P¢) (wo,6A¢) (w0,6L¢)
Oy ~ Pgq T (1~ pso) - - . (3b)
(wo,an) (wO,Pdfi (wo,PdJ)

 

In this notation, for example, 6P represents the change in the neutron
production operator between the final and initial states, P ~=PO,

By restrdicting Eq. (3a) to the special case where the two states
represent critical states (e.g., twe points along the curve o¢ in Fig. 1),

we may set o0, = DSO = 0 to obtain

(Wg,6P9) (¥ ,840) (g, 6L9)
0 = - - : (4)
(bgsPy0) (VP ¢) (U » Py $)

 

This equation is one possible mathematical statement of the reactivity
balance condition as the reactor changes from the initial to final states.
From it, several observalions may be derived that relate to the problems
of constructing a consistent approximation for use in nuclear operations
analysis. First, the changes in the linear operators may be rigorously
separated into a sum of component effects corresponding, for example, to
movements of the control rod, ingrowth of samarium, depletion of uranium,
etc. However, since fthe flux field for the final state involves the cumu-
lative effect of these components, these component reactivities are strict-
ly additive only if it is assumed that the flux distributions in both the
initial and the final staites are known quantities. We will examine the
extent to which this is a real problem later in this section.

A second potential source of difficulty in constructing a comnsistent
approximation for the reactivity balance relates to the multiplicity of
possible paths between initial and final states that may be used to assign
magnitudes to the component reactivities. This may be illustrated by
reference Lo Fig. 1. Consider, for example, two paths between the origin
(point ©) and the state corresponding to the control rod fully inserted
with maximum excess uranium required to maintain criticality (point &).
Let path 1 consist first of the complete insertion of the rod, resulting
in state «, followed by addition of excess uranium to arrive at state c.

For path 2, let the order of these changes be reversed, leading to the
motion obe. Each of these changes belongs to a class for which the change
in the leakage operator 8L can be neglected. (See also the discussion in
the Appendix. This is not equivalent to the statement that the change in
total neutron leakagze is negligible; for the latter, L¢ can differ through

changes in the operand; i.e., the flux distribution.) By use of Eq. (3a),

 

with Pog = Pgp = 0, the motion of the rod in path 1 corresponds to the

static reactivity change
(b,06450.)
P = e, (5)
(0g5Pgt,)

and the compensating reactivity associated with the uranium addition is

(wa,6P¢c) (wa,64U¢@)
., - . (6)
(wa,f0¢c) (wa,P0¢a)

 

In these expressions, GAR and GAU are the changes in the absorption
operator associated with movement of the control rod and additions of
uranium respectively., We have alsec used the fact that PO = Pa’ or
§P = 0 along the path oaq.

For path 2, similar use of Eq. (3b) shows that the static reactivity

change for the excess uranium addition is (P, = P )
b e’ ?

(,,8P6,) (g, 64.56,)
pr == — . : (7)
(yoP o) (WgaP 8,)

 

The compensating reactivity change introduced by the control rod motion is

. T -£E§:ff5fgz-. (8)
(¥, 52,0,)
Comparison of Eq. (5) with (8) and Eq. (6) with (7) shows that, even if we
ignore the differences in the flux fields for the states 0, a, b, e, the
component reactivity magnitudes still differ by the normalization of the

production operator. (In molten-salt reactor applications, the production
10

operator is directly proportional to the uranium concentration.) Moreover,
we can, in principle, develop similar expressions for these components by
using an infinite variety of paths, including that of alternating infini-
tesimal motions in the directions of the coordinate axes, resulting, in
the limit, in a net motion along the curve ¢oc¢ in Fig. 1. In each of these
cases, the reactivity assigned to the total rod motion or uranium addition
will have slightly different magnitude.

As a practical matter, neither of the problems mentioned above pre-
sent serious obstacles in constructing a reactivity balance model. However,
their conceptual recognition is important in developing compatible approxi-
mations for the individual components. For those components that must be
calculated from basic theory, the perturbation approximation for the flux
distribution can be used without seriously restricting the accuracy of the
calculations, as described in Section 2.3. The second difficulty is cir-
cumvented by setting up a specific convention for the reactivity balance
calculations. Care should be taken, however, that this convention also
conforms to the interpretation given to individually measured cowmponents,
particularly the control rod reactivity. These criteria are further dis-

cussed in Section 2.4.

2,2 Simplifications

 

One important attribute of molten-salt reactors that has significant
consequences in constructing an approximate reactivity balance is the per-
sisting uniform distribution throughout the salt of the most important
nuclides influencing the neutronic behavior. This characteristic, more
than any other, makes it practical to develop a model simple enough for
use in on-line computation with a computer serving wany other functions
and still sufficiently realistic for use in long-term monitoring of the
reactor neutronic behavior (without taking recourse to ad hoc renormali-
zations). Specifically, as a first consequence, the changes in nuclide
concentration that occur in the course of operation are governed by the

following expression for the reaction rate:

RALY = N.(8) IVR av JO o, (B) ¢ (1,E,t) F(p) dF , (9)
11

where

Ri(t) = total reaction rate for zth nuclide in salt (events per
second),

Ni(t) = npumber density of Zth nuclide in salt (nuclei ver cubic
centimeter of salt),

Gi(E) = reaction cross section for Zth nuclide (cm?),

$(r,F,t) = neutron flux at point r, emergy F, and time 7 (neutrons

cm™? sec™t),

F(r) = wvolume fraction of salt at point r,

V_ = all reactor volume experiencing neutron flux (cm®).

R

The factoring of the nuclide concentration from the volume integration in
Eq. (9) permits definition of a useful microscopic reactor-average cross
section that is influenced by salt composition, but only indirectly,
through the neutron energy spectrum. In turn, use of these cross sections
simplifies the calculation of the time-dependent nuclide changes and their
associated reactivity effects. These calculations will be described in
Sections 3 and 4 in specific connection with the MSRE reactivity balance
model.

As a second consequence, it becomes possible to factor the concentra-
tions from several of the terms in the reactivity balance and make use of
reactivity coefficients derived either from calculations or from experi-
mental measurements. (An exception to this is the treatment of the reac-
tivity effect associated with the control rods. The evaluation of these
effects is considered in Section 2.4.) To illustrate, consider the theo-
retical variation in the static reactivity as excess uranium is added to
the salt, starting at the minimum critical lecading and maintaining the con-
trol rods withdrawn. This is represented by the curve 00 in Fig. 1. We
will first develop this relation in terms of the general notation of the
preceding reactivity formulas. Following this, the discussion will be
specialized to a particular neutronic model, in order to study the problems
of calculating these coefficients.

By use of the arguments that led to Eq. (7), the static reactivity

for an arbitrary point on curve ob is
12

L Gt g8y

° (BgsP9) (g, Po)

(10

 

Next, we observe that the operators ¢P = P ~»PO and 6AU = AU —-AUO are
macroscopic cross-section-like quantities (i.e., they involve only products
of nuclide concentrations and microscopic cross sections), whose explicit
form depends on the neutronic model. Since the uranium remains uniformly
distributed in the fuel salt and the region geometries and salt volume
fractions remain fixed in variations of the uranium concentration, a
straightforward analysis shows that the concentration may be factored from
these operators. Thus,

T T (11)

and

SA (12)

U

li

(Cy — Cyel¥y o

where CU and O represent the uranium concentration in the salt at an

Uo
arbitrary point along the curve and at the origin, respectively, and H, and
up are microscopic cross—section-like quantities associated with neutron

absorption and production in uranium and also linear operators.

Equation (10) then has the form

0 = 1 m.mmg__g__. e e (13)
(wo,up¢) C

or
c., — C
U uo
K(Cy) G : (14)

©
i

As indicated, the reactivity coefficient, K(CU), depeuds in principle on
the uranium concentration. In application, however, this dependence is

very weak for reasons described in Section 2.3; thus X can be considered
a constant in the calculation of the uranium reactivity component in the

reactivity balance.
13

2.3 Approximate Technigues in Calculation

The simplest theoretical model that includes all the essential fea-
tures of the calculation of neutron flux and reactivity is the two~group
diffusion model. Extension to a larger number of energy groups presents
no difficulty, but the notation tends to be cumbersome and obscure these
egsential features; accordingly, we can discuss the calculation of certain
reactivity effects most clearly in terms of this model.

The correspondence between the preceding general reactivity formulas

and the two-group model is as follows:

¢ P
- tY, weof ! (15)
qb2 wz
Vi Vi,
Do fl fz (16)
0 0
Zal + ?12 0
A - (17)
12 a2
~V-D1V 0
L - . (18)
0 “V-DZV

The direct and adjoint flux vectors are solutions of the equations

. - — - - 7 = 0 1
VeD Vo, =0 Dby ¥ = dvis o) + (L —p)vie ¢, , (19)

al¢1

. — + 7 = 2
VeD Vo, = I b+ T8 0, (20)

VeD V) — 3 0 — % + (1 — VI + 3 = 0, (21)
1 wl alvl 1zw1 ( os) flwl 12w2
14

V'Dszz-— Zazwz + (1 “‘ps)vifzwl = 0, (22)

In these expressions, Dj’ Zaj’ and vffj (7 = 1, 2) represent the diffusion
coefficient, macroscopic absorption cross section, and neutron production
cross section for the "fast" (j = 1) and "thermal" (J = 2) groups respec-—
tively. The slowing-down transfer cross section from the fast-to~thermal
group is 212. In the two~group model, the initial state will appear as a
superscript to aid in separating this designation from the group indices.

The correspondence between the preceding inner product notation and this

model is, for example,

X vl . ¢
1 I 1

(Wo,P0) > [ [0 D) av . (23)
Vg 0 0/ \ ¢,

By applying these correspondences in Eq. (3a) [or (3b)] and carrying
out the matrix multiplications, it is found that the static reactivity
change associated with a change in concentration of a purely neutron-

absorbing material (starting with the state 0) is

2
p—p = . (24)

0 ygz0 0,y dv
IVR (wl YR 4)1 * wl o ¢2)

—_ 0 57 AV ~— 0 sz dv
IVE V%% Y IVR Yy %%, %,

 

Similarly, the corresponding formulas for the reactivity effect of a

change in concentration of fissile material are

1~ 0 gvs + 90 svr, 4 V- 0 51 + 0 g3
¢ °) IVR (wl ¥ f1¢1 wl v fz'z) @ IVR (wl a1¢1 wz 6TCZ2¢2) v

 

0 EO + ¢40 0 v
fvé W2 vrloe + w0 vl e ) d

(25a)

and
15

1 - - (pO8vr + 08yr av - 05z ¢ + %8 dv
L =~0) fv,H(“’l VIp ¢+ l6vIc 8 ) IVRwl B bt 088 8)

 

Oy + pOyuz dv
IVg WPl Yt

(25bh)

If, as in the present application, the change considered is that of a nuclide
uniformly distributed in the fuel salt, it is useful to rewrite these for-
mulas directly in terms of the nuclide concentrations. Setting Fs equal to
the volume fraction of fuel salt in the region exposed to the neutron flux,
GNa and o, equal to the atomic concentrations and microscopic absorption
cross sections of the added absorbing materxial, and N, and vdf equal to the

*

concentration and cross section of fissile material, respectively, we may

rearrange Hq. (24) as follows:

3 (12
o —p = - ’ (26)
o Op + 30 F dv
f IVR(wI Svgfld)l ll11 3v0f2¢2)

s Op dv + Op dav
a (IVR I'.Ul Sgalq)l IVB wZ a d>2 )

 

S g+ g o )
a ¥, ay © 45%,

 

o "'"po = » (27)
NS (F vo,. + vo
P fz)
8N
= K —Z (28)
a NO
7

In the last two forms above, Ka is a reactivity coefficient for the ab-

sorbing material, and the parameters jl and jz are defined by

 

%
We assumed only a single fissile material in writing these formulas:

extensions to account for a mixture are obtained in a straightforward
manner.
F dv
: fvfiw +*)
S (29)
0 av
IVRwl 60,

and

j, = ——, (30)
0 p AV
IVRwI 50,

A similar rearrangement of Eqs. (25a) and (25b) yields the following:

s | - 0y — g i
o [a Gt o) =iy s, |
P P - —E— (31a)
Nf Jivo,. + vo
1 I

 

and

 

p—p = . (315)
0 Nf J vo. + vo

1 1 f2
The second expression above may also be rewritten in a form similar to
Eq. (14); namely,
SN,
p—p, = L% (32)

0 Nf f?

where Kf is a reactivity coefficient for the fissile material.

In all the preceding descriptions, we have taken care to distinguish
between the flux distributions in the initial and final states in a reac-
tivity change. An approximation that affords a great simplification in
the practical calculation with these formulas is the first-order pertur-
bation approximation. Here the flux distribution corresponding to the
final, or perturbed, state is set equal to the distribution in the starting

state before carrying out the necessary integrations to obtain the reac-

tivity coefficients. Although this approximation is often emploved in
17

problems of this general type, the basis of its applicability and its
limitationg are not always discussed. Toward this end, we may first ob-
serve from any of the preceding reactivity formulas that this quantity is
independent of the normalization (magnitude) of either the flux distribu-
tion for the final state or the adjoint distribution for the initial state.
Hence the reactivity calculated with these formulas can conceivably vary
with the final flux distribution only through changes in the relative
spatial or energy distribution of the flux. Second, in the type of reac-
tors we are considering, most of the neutron reactions occur in the energy
region of neutron thermalization and, to a lesser extent, the low-energy,
epithermal region. Expressed in terms of the preceding two-group formulas,
this means that the terms involving the adjoint-weighted thermal reaction
rates tend fto dominate in the evaluation of the reactivity. In this situ-
ation, an early analysis by Wolfe® provides a useful criterion for deter-
mining when the thermal flux distribution can be expected to remain essenti-
ally unchanged during prolonged operation of the reactor. In essence,
Wolfe's treatment shows that the net effect of the perturbation on the re~
actor neutron economy should be small compared with the total neutron leak-
age from the reactor, so only a small change in the flux shape is required
to balance the neutron economy against the perturbation. (This criterion
was certainly fulfilled in the MSRE, as can be shown by analysis of the
neutron balance.) However, the applicability of the perturbation approxi-
mation is actually not as restrictive as this might indicate, particularly
when applied to perturbations distributed over the reactor, such as buynup-
related effects. Here the integral nature of the computation has a ten-
dency to smooth out errors in the flux distribution. As might be expected,
this approximation is generally least accurate in application to & strong
and highly localized perturbation (such as a control rod). For these cases,
more exact flux distributions or results of individual reactivity measure-
ments {(such as described in the following section) should be used.

The incentive for use of the perturbation approximation in the treat-
ment of the calculated reactivity terms is considerable, for it minimizes
the number of direct—-eigenvalue/flux calculations that must be perfbrmed
in the course of reactor operation. The fact that it is an approximation,

however, indicates that circumspection should accompany its use in any
18

particular instance, possibly in the form of point checks with results from
more exact analysis. Reference 7 provides much valuable insight and

background discussion concerning applications of the perturbation technique.

2.4 Reactivity Measurements

 

Jo Section 2.2, mention was made of the unecessity for setting up an
ordering convention for reactivity changes that compensate one another and
for assuring that the interpretation of individual reactivity measurements
conforms to that convention. Of the class of measured components of the
reactivity balance, the control rod reactivity is of major importance, par-
ticularly if the normal mode of power operation involves substantial in-
sertions of the rods (as was the case with the MSRE). The rods represent
strong, localized absorbers, and more elaborate and disproportionate ef-
forts are generally required to calculate their reactivity reliably from
basic thecory; in anv case, a basis of measurement should be available for
comparison.

The convention used in this work is based on the concept that the in-
sertion of the control rods always acts to follow and compensate for any
net excess positive reactivity relative to the origin in Fig. 1. To see
how this may be made to conform to the interpretation of rod calibration
experiments, consider the situation depicted in Fig. 2. Here a typical
period—differential-worth measurement and a rod-drop, integral-worth
measurement are indicated (schematically) for one particular uranium con-
centration., The period measurement determines the slope of the reactivity
versus rod position curve for that uranium coacentration and critical rod
position. 1In Fig. 2 this is indicated by the small triangle starting at
point p in the plane perpendicular to the uranium mass axis. By varying
the uranium content, and with it the initial critical positions for the
period experiments, it may be seen that the slope of the reactivity sur-
face S5 in the neighborhood of the critical line can be obtained.

Similarly, the rod-drop experiment is indicated in Fig. 2 as Lhe tri-
angle ped, where cd represents the total negative reactivity resulting
from rod scram from the initial critical position. It should be noted that

these experiments are by nature limited to the subcritical domain, just as
19

ORNL--DWG 71-4939

PERIOD
MEASUREMEN 7~

  
  
     
   

 

CONTROL. ROD POSITION

 

g
x -
/
/

— aa e g e o

|
o

|

|

|

20D DROP
MEASUREMENT

CRITICALITY LLINE

Fig. 2. Graphical description of rod-calibration measurements.

the positive period measurements apply only to a small region above the

critical line. To gain information about the "supercritical reactivity

curve of, which the rod insertions must
cal line, it is necessary to proceed in
pose, we can make use of the discussion
showed that the result of "translating"

lel to the uranium mass axis introduces

compensate to arrive at the criti-
an indirect manner. For this pur-
in Section 2.2. This discussion

a rod reactivity measurement paral-

a correction factor proportional

to the uranium concentration, together with minor corrections associated
with changes in the flux spectra. Assume now that two rod-drop experiments
have been performed, one with the minimum critical uranium loading in the

reactor to measure the reactivity @b in Fig. 1 and the second at an inter-
mediate uranium loading to measure ¢d as indicated. If we correct the re-
activity measured in the first experiment to the loading conditions at the

time of the second experiment by using the following approximation,

CUD
Ao (U = dp (U ) =5 , (33)
'Oab ab 0 CU

it may be seen that the AD;E will be approximately equal to the reactivity

difference between points f and d in Fig. 2. Hence,

ho o = (34)

o

/ —
&pab Ao d

-
20

In this indirect manner, the curve oh can be generated from a number of
rod-drop experiments at various uranium concentrations. From this an
experimental value for the uranium concentration coefficient of reactivity
can be derived [Eq. (14)]. A similar and obvious interpolation procedure
can be applied to the results of period measurements to obtain an approxi-
mate differential-werth curve for a specified uranium loading.

A final topic that should be mentioned before concluding this dis-
cussion of the foundations of the reactivity balance concerns the role of
the effective delayed~neutron fraction in reactivity measurements. The
general theoretical basis for amalyzing these measurements is described in
several textbooks dealing with reactor kinetics (see, e.g., Ref. 8). It
will suffice to observe here that the measurements considered in the pre-
ceding discussion are relative in character, and their interpretation de-
pends on the nature and dynamic behavior of the delayed-neutron precursors.
This can give rise to a source of inconsistency in the reactivifty balance.
In particular, the conversion of the results of measurement to an absolute
reactivity scale, in the sense used in this section, reguires knowledge of
the delayed-neutron source strength relative to that of prompt neutrons
from fission in contribution to the neutron cycle. In addition, another
characteristic of delayed neutrons frhat must be taken into account is their
energies of emission, which are lower than those of the prompt neutrons
from fission. In a thermal reactor such as the MSRE (which had a high
fraction of neutron leakage to the surroundings), the net result of this
difference in emission energy spectra is that the delayed neutrons have
less likelihood of leakage than the prompt neutrons, so their contribution
to the chain reaction is enhanced. Hence, both theoretical analysis and
physical intuition lead to a "delayed-neutron effectiveness' factor Y by
which the delay fraction Bi for the ith precursor group must be multiplied
to obtain their contribution, relative to the prompt neutrons, in promoting
the chain reaction.

Chapter 6 of Ref. 8 contains an extensive discussion of the problem
of calculating the delayed-neutron ecffectiveness factors. Although several
measurements of the delayed-neutron emission spectra have been made, suf-
ficient uncertainties still exist in the data to suggest that detailed

calculations to take into account the shape of the delayed-neutron energy
21

spectra are not always warrvanted. For thermal reactors in particular, an
adequate estimate of the corrections arising frowm these differences in
energy spectra may be obtained by assuming that all delayved neutrons are
emitted with a single average energy, independent of the precursor group.
According to data taken from Ref. 8, this average energy is approximately

0.43 MeV, Then Y, ® ;-can be cowmputed from the approximate formula

_ +B2(TD-T5) . (35)
Yy = e o

in which B® is the geometric buckling corresponding to the "nuclear" size
of the core and Tp and T, are the values of the average age to thermal

energy for prompt fission neutrons and for delayed neutrons respectively.

These effective delayed-neutron fractions, Eé = ;Bi, with no circu-
lation of the fuel, form the basis for normalization of the "measured" re-
activity components, Details of the method of numerical evaluation of‘?
for the MSRE are given later in this report (Sect. 4.5).

The procedure described above is strictly sufficient only for reacfors
with solid fuel and fluid-fuel reactors with negligible circulation of the
precursors., For the MSRE the calibration of the countrol rods and the re-
activity measurements derived frow these experiments were performed with
the circulation stopped. When the fuel is circulating, additional com-
plexities arise from interaction between the fluid flow and precursor dy-
namics. This delayed-neutron loss is best treated as a separate (negative)
reactivity component effect that can be precisely measured and normalized
on the basis described above. For conditions of steady~state circulation
of fuel, this effect is constant and therefore equivalent to a zero-point
correction in the reactivity balance.

The theoretical and experimental interpretation of the delayed-neutron
loss effect in the MSRE, Dboth under steady-state and transient-stable

period conditions, was the subject of an earlier report.9
22

3. DESCRIPTION OF INDIVIDUAL REACTIVITY EFFECTS IN THE MSRE

3.1 Reference Conditions

The discussion given in the remainder of this report applies specifi-
cally to the MSRE. Some knowledge of the basic reactor design character-
istics is assumed, and further details are available in Ref. 10,

If changes in component reactivity effects during operation are to be
monitored, it is advantageous to choose a starting, or reference, condition
that can be defined by experimental measurements with relatively little
error or ambiguity. The reference conditions chosen for this work corre-
spond to the just—critical reactor, isothermal at 1200°F by use of the ex-
ternal electric heaters, fuel circulating and free of fission products, and
all three control rods withdrawn to their upper limits (51 in.). The
uranium concentration for these conditions, as well as the increase in
uranium concentration required to compensate for a range of control-rod
insertions and isothermal temperature changes, was established during a
program of zero-power nucleac experiments. The experimental program per-
tinent to the first uranium 1oading,11 a mixture of 235U and 238U, were
carried out in the summer of 1965. 1In this program, independent measure-~
ments of the control-rod reactivity worth (period—differential-worth
experiments and rod-drop integral-worth experiments) were used to determine
reactivity equivalents of uranium concentration changes and isothermal
temperature variations.

The zero-~power experiments pertinent to loading with 233U were carried
out in the fall of 1968. This program established the necessary base-line
conditions and information for reactivity balance calculations during power
operation with 233y, A summary of the results was given in Ref. 12, and
further details will be included in subsequent topical reports concerned

with analysis of nuclear operation with 233y,
3.2 "The General Reactivity Balauvce Equation

The equation describing the general situation when the reactor is

operating at some specified power level includes terms representing,
23

relative to the reference state, (1) the positive effect of the total ex-
cess uranium added before increasing the power, (2) the poisoning effect of
the rod insertion, and (3) the power and time-integrated power-dependent

effects associated with changes in fuel and graphite temperature levels and

distributions, 233y burnup, and fission product buildup (135Xe, 149Sm,
151gm and nonsaturating fission products). These terms include the most
important direct effects of substantial power generation. There are, how-
ever, other indirect effects arising from isotopic changes. These include,
(1) the burpout of the small amount of 5Li present in the clean fuel salt,
(2) burnout of residual 9B from the unirradiated graphite moderator,

(3) preduction of plutonium from absorptions in 238U, and (4) changes in
the concentrations of the other nonfissile isotopes of uranium during opera-
tion. There are, in additiom, othér known reactivity effects that can be
shown to be insignificant in the MSRE, such as photoneutron reactions in
the beryllium in the fuel salt, changes in rod worth due to gadolinium de-
pletion, and several high-energy neutron reactions. These complete the
list of component reactivity effects only if we assume that the structural
configuration of the graphite stringers and the associated matrix of fuel-
salt channels underwent no significant changes during the power-generating
history of the core. If changes in the fuel-moderator geometry were in-
duced, for example, by nonuniform temperature-expansion effects or cumu-
lative radiation-damage effects on the graphite, this could appear as an
anomalous reactivity effect not explicitly accounted for in the reactivity
balance. Theoretical calculations indicate that some radiation-induced
geometric changes may have occurred; these calculations and their influence
on the interpretation of reactivity balance data are described in

Section 3.3.

There is substantial evidence that another special reactivity effect
was of importance in the operation of the MSRE., This arose from the en-
tvainment of helium-gas bubbles in the circulating fuel salt through the
action of the xenon-stripping spray ring in the fuel-pump tank. These
minute, circulating helium bubbles would be expected to affect the reac-
tivity in two wavs: (1) by modifying the neutron leakage through an ef-

fective reduction in the density of the fuel salt and (2) by providing an
24

additional sink for !3%°Xe and thereby reducing the effective xenon migration
to the graphite pores. (These effects will be discussed in greater detail
in a later section.)

We can summarize the preceding discussion in the form of a general
equation for the reactivity balance. By using the symbol Ap(xr) to repre-~
sent the algebraic value of the reactivity change due to component x, and
grouping terms that can be treated similarly in the calculations, we

obtain

0 = Ap (rods) + Ap (excess 23%U) + Ap (temperature) + Ap (power)
+ Ap (Sm) 4+ Ap (135Xe) + Ap (other nuclide transmutation effects)
+ Ap (residual), (36)

The final term on the right-hand side of this equation includes effects
divisible into three basic categories: (1) any residual effects known to
normally occur in the reactor core that are not explicitly accounted for
in the calculations, (2) residual reactivity corrections due to any errors
in calculating the other terms, and (3) changes that could be considered
to constitute anomalous behavior (such as uranium separation from the cir~
culating fuel salt).

Figure 3 illustrates diagrammatically how the information required
for on-line calculation of the various terms of Eq. (36) was assembled.
In the figure the solid arrows are used to designate the "primary' sources
of information used to evaluate the reactivity effects. The dashed arrows
indicate where the dependency on theoretical modeling is secondary. This
produces a rough separation of the terms into two groups (emphasized by
the double line), according to the extent to which experimental measure-
ments were sufficient to evaluate these terms. The dependency of evalu-
ations of the top group on theoretical input resides mainly in the evalu-
ation of the effective delayed-neutron fractions required to convert the
results of the rod-calibration experiments to an "absolute" reactivity
scale.

The bases of evaluation of the terms of Eq. (36) are described in
greater detail in the remainder of this section in the order listed in

the equation.
25

ORNL-DWG 71-6286
REACTIVITY EFFECTS:

 

 

ZERO-POWER NUCLEAR

EXPRRTMENTS CONTROL RODS

 

 

 

 

™ OUTLET TEMPERATURE

 

 

MEASURFMENTS IN EARLY /I TEMPERATURE DISTRIBUTION

POWER OPERATION

 

| (Pover -Reactivity

 

 

 

(Observed Conmtrol-Rod Coefficient) TN STRUMENT
Compensations) A MONITCRS:
| 2357 CONCENTRATION e e e ]

 

| (Depletion and Additions
EP ) (Power level,

temperatures, rod
. positions, operating
+35¥e POLSONING time)

 

 

INITIAL CORE
COMPOSTITLON

i

THECRETICAL MODELING:

 

 

 

 

 

 

 

14‘35“1 + lSlSm
POLISONING

 

 

Average Reaction Cross
Sectiouns, Fluxes,
Renctivity Coefficients,

OTHER NUCLIDE
TRANSMUTATTON

 

 

 

Effective delayed- (°ri, Cp, =5% + 238y,

neutron fractions, Mass- 239py + 2%y, Non-

transport effTects Saturating Fission
Products)

 

 

 

 

RESIDUAL REACTIVITY

 

 

 

Fig. 3. Composition of MSRE reactivity balance.

Control-~rod reactivity worth

 

Quantitative determinations of the rod worth, the 235U reactivity
worth, and the temperature reactivity effects were made during the zero-
power experiments., Because the uranium and temperature reactivity effects
are inferred from the control~rod calibration experiments, and also because
the magnitude of other known power—dependent reactivity effects, described
later, are evaluated empirically according to the changes in control-rod
position following a change in power level, precise determination of rod

worth is wvital to the successful interpretation of the reactivity balance.
26

The control rods were calibrated by means of rod bump—period measure-
ments made with the reactor at zero power (i.e,, with negligible tempera-
ture feedback effects) and the fuel circulating pump stopped11 during a
period of uranium additions sufficient to vary the initial critical posi-
tion of one rod (the regulating rod) over its entire length of travel. At
three intermediate #3°U concentrations, banked insertions of the two shim
rods required to balance specified increments of withdrawal of the regu-
lating rod were measured. In this way, various combinations of shim-~ and
regulating—-rod insertions equivalent in their reactivity poisoning effect
were obtained. Rod-drop experiments were also performed at three inter-
mediate 235U concentrations. In these experiments, the equivalent integral
negative reactivity insertion of the rod in falling from its initial criti-
cal position to its scram position was measured. Agreement between the
integral of the differential-worth measurements and the integral reactivity
obtained directly from the rod-drop experiments was found to be within 5%.
The reactivity versus position calibration curve for the regulating
rod, and the results of the three experiments measuring equivalent shim-
and regulating-rod combinations were next combined with an approximate
theoretical formula for the reactivity worth of an arbitrary shim—regu~
lating-rod configuration. The theoretical formula contained several parame-
ters that were adjusted so that the formula provided a least-squares fit
to the experimental measurements, (Derivation of the formula for rod worth
and discussion of its application are given in Ref. 13.) The result of
this analysis is shown in Fig. 4. The solid points are data taken from
smooth curves drawn through the experimental data and are normalized to
the uranium concentration attained at the end of the zevo-power experiments
with 235U, (As noted in Sect. 2.2, the rod reactivity worth is inversely
proportional to the 235y concentration within its range of variation.)
Figure 4 indicates that the smoothed data could be fitted very closely with
the theoretical rod~worth formula, except for small evrors at the extreme
positions of the rods (full insertion or withdrawal). No important re-
strictions in the use of the formula arose from these errors, since the
purpose was primarily interpolation of the reactivity worth of intermediate
shim~regulating-rod combinations not specifically covered in the three

groups of experiments described above. The formula provided a convenient
27

ORNL-DWG 56-4758

 

 

 

 

  
  
 
 
 

 

 

 

 

 

 

 

      

     

 

 

 

   

 

 

 

 

 

 

 

 

 

 

T
2.0 | _ -
® POINTS OBTAINED FROM ROD-CALIBRATION
EXPERIMENTS

1.8 |— —— CALCULATED FROM LEAST-SQUARES
< FORMULA
T 6l {ROD REACTIVITY NORMALIZED TO |
< 74.74 kg 23U IN LOOP)
52 <
— 1.4 /
T
|_
S
212 —
>
=
> 1.0 ———
=
[
N ‘
gog il b
d
<
T 06 I
Ll
’_
z

0.4 +—--m e

0. 2 XA - _ e S

2 L/-/’ ‘o/ c"o/ C;z?“é- 3
9 i
o L& . A A ‘
0 4 8 12 16 20 24 28 32 36 40 44 48 52

POSITION OF ROD MOVED (in. WITHDRAWN )

Fig. 4. Comparison of control rod reactivity from experimental
curves and from the least-squares formula.

means of rapidly calculating the reactivity equivalent of the rod config-
uration during reactor operation by means of the on-line computer. One
potential restriction in the practical use of the formula should be noted,
however. It could only be applied in regions of rod travel and excess re-
activity covered in the zero-power calibration experiments (i.e., magnitude
of reactivity less than or equal to the worth of a single rod moving through
51 in. of travel). Modifications of the least-squares formula would have
been required to cover a larger reactivity range. Since the reactor fuel
loading never exceeded that attained in the initial calibration, this
limitation did not affect the actual operation.

Checks were made during power operation of the reactor (see also the
discussion in Chap. 3) to deterwmine whethey use of the approximate formula
would lead to any significant error in the reactivity effect assigned to
the rod. These were done by varying the shim~ and regulating-rod configu-
ration required to maintain constant reactor operating conditions and com-

paring the calculated reactivity effects. These tests indicated that the
28

maximum reactivity variation attributable to systematic errors of this
type would be about 0.03% §k/k and would generally be associated with the
largest changes in shim- and regulating-rod configuration occurring during

routine power operations.

Excess 23°U reactivity worth

Relative to the reference conditions defined in the preceding part of
this report, the total excess 2357 is equal to the amount added during the
zero-power experiments (in the form of highly enriched uranium) minus the
amount burned during power operation of the reactor plus the amount added
to reenrich the fuel salt when the burnup becomes sufficient. Corrections
must also be introduced for relative dilution effects each time the reactor
fuel loop was drained and the fuel mixed with the fuel salt '"heel" re-
maining in the drain tanks during operation, as well as for absolute dilu-
tions* from reactor flushing operations.

The reactivity equivalent of the excess uranium was determined from
the zero-power experiments by measuring the amount of control~rod insertion
required to balance each addition of 235 and then using the independent
calibration of reactivity versus position to determine the incremental re-
activity worth of the 235U, The theoretical foundations of this measure-
ment were described in Chapter 2. It was shown that the form of the ex-
pression for the excess uranium reactivity effect was

K(C — C,)
Ao (excess 23°1)) = —m—— (37)

C
where C is the concentration of 235U in the salt, 'y is the value at the
minimum critical loading, and K is approximately constant over the range
of concentrations encountered in operations. The parameter K is also equal
to the concentration coefficient of reactivity at the reference conditions,
as can be shown by differentiating Eq. (37) and setting the concentration

equal to that at the minimum critical loading:

 

A nominally uranium~free flush salt was normally used to rinse the
primary system prior to and after operations that involved opening the
system (e.g., replacement of core irvadiation specimens). TFlush salt left
in the primary loop after draining caused the fuel dilution.
29

d Ap) -k
o _ ok (38)
( AW Je=c,

Results of fitting the data obtained during the zereo~-power experiments with
an expression of the form of Eq. (37) are shown in Fig. 5. From this anal-
ysis, a value of K = 0.24 was obtained; correspondingly, the average con-

centration coefficient of reactivity over the total variation in 2357

loading during the zero-power experiments was 0,22,

In order to evaluate the changes in concentration due to 235U de-
pletion in power operation, theoretical calculations of the fission and
radiative capture cross sections, averaged over the reactor spectrum, were
required. These calculations are described in Chapter 4. The analysis
indicated that the 23U would be consumed at the rate of 1.31 g/MWd, or
at the equivalent rate of about 5% of the circulating inventory per full-

power year.,
Temperature reactivity effect

When the core temperature is maintained spatially uniform, a change

in this temperature can be related both experimentally and theoretically

ORNL-DWG 71-4940

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DD e
e DATA CORRESPONDING TO VARIOUS CAPSULE _
50 ADDITIONS, DETERMINED FROM INTEGRATION -~ A
OF PERIOD MEASUREMENTS. FUEL CIRCULATION -]
F 18 STOPPED /
o s
© 16 THEORETICAL CURVE FIT ./ ,,,,,,,,,,,,,,,,,,,,,,,,,,,,
2 ~
>...
I_—..:
>
I.__.
)
<
L
(e
N
L 06 e "m..............J./.j......-.....7.*7.‘..__ .......................................... .,.,.....____.W---....J‘i ........ _ ]
< ' i‘
0 Q4 [ g ]
-l‘/
02 {3 . -
O .. 1“/.’ J A
65 66 67 68 69 70 71 72

MASS OF 235 IN FUEL LOOP {kqg)

Fig. 5. Effect of 2330 mass on reactivity in first MSRE loading;
theoretical curve fit.
30

to the core reactivity in an unambiguous manner. The method used to
measure the isorhermal temperature coefficient of reactivity during the
zero-power experiments consisted in varying the external heater inputs and
allowing the just-critical reactor to cool slowly and uniformly while
measuring the change in regulating-rod position required to maintain a
constant neutron level, 1In these experiments the fuel was circulating and
the system temperature was taken to be the average of the measurements made
with preselected thermocouples distributed over the circulating system.

The change in rod position corresponding to the temperature change was con—
verted to reactivity by again using the rod calibration curve.* The zero-
power experiments measured the combined effect of a uniform change in fuel
and graphite temperature, The value of the total isothermal temperature

coefficient of reactivity obtained by this means was —/.3 x 1075 °fF~l,

Power coefficient of reactivity

 

At power levels higher than about 10 kW of heat, spatial distribu-
tions of temperature due to the nonuniform heating of the core begin to
become significant. The reactivity change, relative to a fixed uniform
temperature level, is no longer simply related to a single physically
measurable temperature (or even the average of several measured tempera-
tures) in the circulating system. Rather, the reactivity is a cumulative
effect of the entire temperature field in the core. This temperature-
distribution reactivity effect, or steady-state power coefficient of re~
::1c1:1',\n',ty,.r is somewhat difficult to estimate reliably for the MSRE because
it requires accurate knowledge cf the local heat deposition and tempera-
ture distributions in the graphite and salt and the contribution of these
local effects to the neutron reaction rates. An approximate way of treat-

ing this problem involves the use of a "nuclear average temperature,'" as

 

*
Interaction effects (i.e., effects of the temperature change on the

total rod worth) were estimated from theoretical considerations to be
quite small.

T

The temperature distributions in fuel and graphite are determined by
the total power level and the mode of temperature level control (the reac~
tor outlet fuel temperature was servo-controlled in the MSRE). Since the
power level was an input variable to the on-line computer, it was con+-
venient to relate the reactivity effect directly to the power level.
31

described in Ref, 14. 1In this method, the local temperature changes are
multiplied by a weighting (importance) function that measures their effect
on the net reactivity and then integrated over the reactor core. Even if
we assume that the temperature distributions io the fuel and core graphite
can be calculated accurately, it should be noted that the weighting pro-
cedure described in Ref. 14 is theoretically insufficient when applied to
a small reactor core, such as the MSRE. Here the principal temperature
reactivity effects arise from changes in the neutron leakage. Although
nonuniform temperature changes induce expansion in fuel salt and graphite
that affect the reactivity according to the weighting procedure indicated
above, the complete description of the effects would also require calcu-
lation of neutron thermalization in a nonuniform temperature field. Dif-
ficulties in the practical computation of these effects can be a source

of error in theoretical evaluations of the power coefficient.

The power coetficient of reactivity for a fixed reactor outlet tem—
perature was measured during the approach to power by holding the reactor
cutlet temperature at a preset value with the servo controller and meas—
uring the control-rod response to the change in steady-state power level.
Since the reactivity response to the change in temperature distribution is
essentially instantaneous, this effect can be separated from the slower
power—dependent reactivity effects, such as the !35Xe and '%%Sm effects.
The total effect was quite small, and the measured power coefficient was
+0.0017% reactivity per megawatt. This observed coefficient corresponded
to a difference of about 3°F/MW between the nuclear average temperature of
the graphite and that of the fuel. The measured coefficient was applied

to evaluate the term Ap (power) in Eq. (36).

Samarium poisoning

 

The term in Eq. (36) representing the poisoning effect of 1"%%Sm and
151gm cannot be conveniently separated in experimental measurements and is
best calculated from basic theoretical considerations. The direct fission
production-decay schemes for these high-cross—section nuclides are shown
in Fig. 6. The parameters 3a¢ represent the reaction rate constants for
neutron absorption, normalized to unit power level and corrected for the

time the fuel spends in the part of the circulating loop external to the
32

ORNL—DWG 71-—9806

149 CHAIN 151 CHAIN
YIELD: 1.13% YIELD: 0.44%
149PITI 151Pm
X (0.0130/hr) x(0.0248/hr)
149Sm_.....m—-,_w‘|5osm 151SmM1525m
49 51
749 ¢ 51 ¢
SAMARIUM-—-149 SAMARIUM~ 151

Fig. 6. First-order decay schemes for production of samarium poisons.

reactor core and thus out of the neutron flux., The calculation of the flux
and =ffective reaction cross sections for the samarium and other nuclides
is described in a later section of this report.

In principle, the chains shown in Fig. 6 should be connected by neu-
tron absorption in 15OSm; other indirect routes for the production of 1493,
can also be considered. However, for the relatively low neutron flux and
fraction of uranium burnup engendered in the MSRE, these corrections can
be neglected. For periodic calculation with the on-line computer, the
differential equations describing the production and decay schemes in
Fig. 6 were converted to finite difference form. The form of the equations

used for computation in both decay chains was

Ny (t, + 88) = U (L) — X ALY + CQ(E) AL (39)
N (L. +88) = N (£ )1 =0 9Q(E,) bt] + N (E.)) At (40)

_ At + A, + 61)
Q) = 5 , (41)

 

where N(ti) is the atomic concentration of the isotope in the fuel salt at

time ti’ subscripts P and S refer to promethium and samarium, A¢ is the
33

time interval between calculations of the concentrations, and § is the
average power level during this time interval. The coefficient Cl is the
lproduct of the direct fission yield and the average fission rate per unit
volume of fuel salt normalized to 1 MW.

Conversion of the 1“%Sm and !51Sm concentrations to reactivity effects
required calculations of the reactivity coefficients for unit absorbers of
this type uniformly distributed in the fuel salt. The theory for these
calculations was described in Section 2.3 of this report: specific details
on the MSRE nuclear computations are given in Chapter 4.

For the specific example of a step increase in reactor power from zero
to 7.25 MW (approximately the maximum heat output of the MSRE), the magni-
tudes of the samarium reactivity effects are given in Fig. 7. Although
these indicate the general magnitudes and time constants, it should be
emphasized that these magnitudes apply to this simplified power~time his-
tory only; the magnitudes in the reactivity balance calculations differed

somewhat, depending on the actual power—-time history in reactor operations,

ORNL-DWG 7{-4941

TOTAL SAMARIU

i —
o —

SAMARIUM-149

SAMARIUM--151

REACTIVITY MAGNITUDE (% 8k/k)

 

O 10 20 30 40 50 0 70 B8G

TIME~INTEGRATED POWER (1012 MWhr)

Fig. 7. Samarium poisoning reactivity effect in MSRE for a step
increase from 0 to 7.25 MW at start of power operation.
34

135%e poisoning

 

Prior to nuclear operation of the MSRE, estimates of the magnitude
of 133%e poisoning were based on the assumption that at equilibrium a rela-
tively large fraction of the xenon produced in the reactor would diffuse
into the pores of the graphite moderator and undergo radioactive decay and
neutron absorption there. Continuous removal of some of the xenon from
the fuel salt would be accomplished by circulation of a small bypass stream
of salt through the spray ring in the fuel-pump tank, which contacted the
salt with a stream of helium gas. Estimates of ihe efficiency of removal
of fission gases by this stripping apparatus and also of the expected mass
transfer of xenon to the graphite pores were based on experiments with
85k tracer performed prior to nuclear operation of rhe MSRE. From these
experiments a model was developed to describe the xenon behavior in the

reactor.15

It was recognized early that the presence of any circulating
voids (undissolved helium gas) could drastically affect the xenon behavior,
and these effects were considered in the model ultimately developed in

Ref, 15. The circulating gas bubbles are effective in reducing the poison
level due to the combined effects of the large overall gas~liquid inter-
facial area for mass transfer to the bubbles and the large xenon storage
capability of the bubbles (because of the extreme insolubility of xenon in
molten salt). Thus not only do the bubbles compete effectively with the
graphite pores for remeoval of xenon from the liquid, but 135Xe in the cir-
culating fluid is a less effective poison than that in the graphite because
about two-thirds of the fluid is outside the core at any instant.

When power operation of the MSRE began, the first attempt at on—line
calculation of the 13°Xe poisoning term in the reactivity balance did not
account for the effects of circulating bubbles. Indications from densi-
tometer measurements during prenuclear testing and experiments during
initial nuclear operation had been that practically no bubbles would
circulate with the fuel, Also, in the early experiments, several tests
had been performed to evaluate the response of the reactivity to changes

in system overpressure.ll

The system pressure was slowly increased by
about 10 psi and then rapidly reduced to the normal value. If circulating

bubbles had been present, their expansion when the pressure was reduced
35

would have expelled some salt from the core and reduced the nuclear reac-
tivity. In addition, the gas expansion in the entire loop would have
raised the salt level in the fuel-pump tank. There was no evidence of
undissolved gas in the tests performed with the normal salt level in the
pump tank. However, when the salt level was reduced to an abnormally low
value, the same experiments did indicate some undissolved gas. Since this
gas appeared only under off-design conditions, we concluded that circulat-
ing bubbles would not be a factor in the xenon poisoning during normal
power operation.

Soon after power operation of the reactor was started, it became ap-
parent that the magnitude of the steady-state 135%e poisoning was much
smaller than had been predicted on the basis of the above considerations.
At this point the attempts at on-line calculation of the xenon poisoning
were suspended, and the reactivity-~balance results were used to measure
the actual xenon poisoning. Examination of the steadv-state results
showed that the low poison level could not be accounted for with reason-
able parameter values within the assumption of no circulating bhubbles.

In addition the system response to small pressure changes now indicated a
small circulating void fraction at normal salt levels in the pump tank.

In view of new evidence for circulating bubbles, the steadyv-state

effect was reevaluated. As expected, the steady-~state xXenon poisoning was
quite sensitive to both the volumetric void fraction and the bubble-
stripping efficiency, decreasing monotonically with increases in either

of these parameters. However, it was found that the steady-state xénon
voisoning as a function of reactor power could be described by a variety
of combinations of void fraction and bubble~stripping efficiency. There~-
fore the analysis was extended to include the time dependence that would
permit a comparison of calculated and observed transient 13%%e poisoning
effects (as determined by the change in the critical position of the regu-
lating rod during the first 40 hr following a change in the steady-state
reactor power level), The purpose was to attempt a separation of those
parameter effects that could not be separated in the steady-state corre-~
lations. The mathematical model used to calculate the time behavior of
the 135%Xe poisoning is described in Ref. 16. Here we will give only a

brief description of the main aspects and assumptions of the calculation.
36

The model chosen was patterned after that developed in Ref, 15. We
assumed that all the 1351 produced from fission would remain in circulation
with the salt and that after decay to 135Xe, the xenon would migrate to the
accessible pores of the graphite at the boundaries of the fuel channels
and also to minute helium bubbles distributed throughout the circulating
salt stream. An effective mass~transfer coefficient was used to describe
the transfer of xenon from solution in the circulating salt to the inter-
face between the liquid and the graphite pores at the channel boundaries.
Equilibrium Henry's~law coefficients were used for the mass transfer of
xenon between the liquid phase at the interface and the gas phiase in the
graphite pores. The numerical value used for the mass—-transfer coefficient
between the circulating salt and the graphite was based on the krypton-
injection experiments with flush salt circulating in the fuel loop.

Similar assumptions were made vegarding the mass transfer of xenon
from liquid solution to the gas bubbles. The coefficient of mass transfer
from the liquid to a small gas bubble of the order of 0.010 in. in diameter
moving through the main part of a fuel channel was estimated from theo-
retical mass~-transfer correlations.'® The equilibrium 135%e poisoning was
shown to be relatively insensitive to the average bubble diameter and mass-—
transfer coefficient over a reascnable range of variation (uncertainty) for
these parameters.

Different efficiencies of removal by the extermal stripping apparatus
of xenon dissolved in the salt and that contained in the gas bubbles were
provided for in the computational model. The efficiency of removal of
xenon dissolved in the salt (fraction of xenon removed per unit circulated
through the spray ring) was estimated to be between 10 and 15%, based on
some earlyv mockup experiments for evaluating the performance of the xenon
removal apparatus. By contrast, the efficiency of removal from the gas
bubbles could be considerably higher, depending on the probability of re-
placement by fresh sweep-gas bubbles in passage through the spray ring
into the pump~bowl reservoir.

The conversion of the calculated '3%fe concentrations in salt, gas
bubbles, and graphite pores to the corresponding reactivity poisoning ef-
fect followed from considerations similar to those described in the pre-

ceding section for the samarium isotopes. Here, however, there was one
37

special feature that had to be accounted for which is not present in the
case 0of samarium. This arose from the nonuniformity of the spatial
distribution of the 13%Xe in the graphite pores. For the graphite region,
theoretical analysis indicated that the 135%e would tend to assume a char-
acteristic shape governed by the burnout of the xenon in the neutron flux.
The concentration would be lowest near the center of the reactor and high-
est near the boundaries of the graphite region. This influenced the net
reactivity effect, since these regions assume different importances in de-
termining reactivity changes, and therefore a '"shape-correction factor"
for this effect was required.

To develop an approximate model for on-line calculations of the 135xe
effect, a computational study based on the theoretical model described
above was first performed "off-line,” with the aid of an IBM 7090 program.
These theoretical calculations were compared with the residual reactivity
data obtained from on-line calculations by using Eq. (36), with Ap (135Xe)
set equal to zero. The apparent transient 135%e poisoning was determined
by subtracting all other known power-dependent reactivity effects from the
reactivity change rvepresented by movement of the regulating rod during
the first 40 hr after a step change in the power level, This off-line
analysis was the most efficient method of making a first~vound analysis
of the transient 13°Xe behavior because the many other usage requirements
of the data logger limited us to a relatively simple "point" kinetic model
for on-line computations and also because a wider parameter study could
best be performed on a larger machine,

A detailed account of the results of comparing the calculated be-
havior with reactivity transients observed during the first few power runs
of the MSRE was given in Ref. 17. This analysis showed that while the
apparent !35Xe poisoning at steady state could be explained by a large void
fraction (between 0.5 and 1.0 vol %) and a low bubble stripping efficiency
(~v10%), the transient behavior could not be closely fitted using these
assumptions. The opposite assumption of a relatively high stripping ef-
ficiency (Eb) and lower bubble fraction (qb) not only fitted the 135%e
transients better but was also consistent with the rates of excess gas
removal observed in pressure release experiments, Figures 8 and 9 are

representative of the data comparisons obtained for a step increase and
38

) ORNL—-DOWG 67—-41080
T T T

i ‘ | ‘: | i

; 1 1 ! T

S | ; | ! ¢, BUBBLE STRIPPING ———

1 | | EFFICIENCY (%)
e EXPERIMENTAL DATA, OSTAINED |
__ DURING MSRE RUN NO.8 ——.

0.7

 

 

O
™

 

  
   
    
 

£ (T Sk/k)
o
n

o
M

FIVITY MAGNITUD
o
]

REACT
Q
!

 

 

 

 

 

O 4 8 12 15 20 24 28 32 36 40
TIME AFTER INCREASE IN REACTOR POWER LEVEL (hr)

Fig. 8. Effect of bubble-stripping efficiency on transient buildup
of !3%Xe reactivity. Step increase in power level from 0 to 5.7 MW;
volume percent circulating bubbles, 0.10; MSRE run 8.

ORNL-DWG 67-1087

 

 

 

 

 

0.7
[ |
\
< 06 /41&— ------ ® EXPERIMENTAL DATA  OBTAINED — —
> “‘\\\\\ DURING MSRE RUN NO. 10
K
& o5 ™, |

i
=

0.4 7szmmm"“*\_m \\\\\\\ ; 1 1

 

| |
¢, .BUBBLE STRIPPING

 

 

REACTIVITY MAGNITUD

 

 

 

 

 

 

 

 

 

 

 

 

 

 

03 — - P, - EFFICIENCY (%) -
,g"”é' “'0'@»—&_“_ - \
L-® @ G}E‘ 10
2 h .
e 7’_@%{:‘;&\\ g . \ - i_ ------ N
~ *3 20
[ \0\\ |
\‘ B 0,‘_3 50 \\N
£ | = "8 @: e
O ‘-“"‘-¢-fi—'““___;_ w?“"p"g'-_.e__ ‘*w.,._bj
: T~ . L 8
J 100 $—0 81y
. . | ]
0 4 8 12 16 20 24 28 32 36 40

TIME AFTER DECREASE IN REACTOR POWER LEVEL (hr)

Fig. 9. Effect of bubble-stripping efficiency on transient decay of
135%¢ reactivity. Step decrease in power level from full power to zero;
volume percent circulating bubbles, 0.15; MSRE run 10.
39

a step decrease in the reactor power respectively. The calculated curves

for the reduction in power level (Fig. 9) also reveal an important char-
acteristic of the tramnsient xenon hehavior that is due to variations in
the overall xenon distribution resulting from choice of the parameters

a and £ e If the circulating void fraction is relatively low, most of
the steady-state poisoning effect is due to 135%e in the graphite, and
only a small amount of xXenon is in the circulating fluid. ZXenon that is
produced in the fluid from iodine decay continues to migrate to the gra-
phite for a period of time after the power has been reduced, This produces
a shutdown peak in the =xenon poisoning. Eventually, the stripping process
reduces the xenon concentration in the fluid so that some of the xenon in
the graphite can escape and be stripped out. This results in a more rapid
decrease in xenon poisoning than simple radioactive decay. If either the
bubble stripping efficiency or the circulating void fraction is increased
(the latter effect is not shown explicitly in Fig. 9), the bubbles become
a more effective sink for the additional xenon produced from iodine decay,
and there is less xenon migration to the graphite. In this case the shut-
down peak tends to disappear. This effect makes the shape of the shutdown
transients more sensitive to changes in the values assumed for the bubble
parameters and thus facilitates the process of fitting the observed data
to the calculations.

The parameter studies illustrated by the examples in Figs. 8 and 9
indicated that the circulating void fraction and bubble stripping effici-
ency might be bracketed between 0.1 and 0.15 vol Z and 50 to 1007 re~-
spectively. Within these ranges, the transient model appeared reasonably
insensitive to the choice of parameters. A combination of oy = 0.15 vol %
and £ = 50%Z was chosen for further application in the on-line calcula-
tions,

Based on the results of the off-line analysis described above, approxi-
mate equations and parameters were determined for nominal on~line calcula-
tions of the !3%Xe reactivity effect during the remainder of power opera-
tions with 235U, Similar to the case of the samarium poisoning calculation,

these were finite difference equations of the form given helow:
40

1135(151: + AE) = :[135(752:) (1 = a, At) + alé(ti) At (42)
135 — 135 — " -y
XS (ti + AE) X8 (ti) [1 a, At aaQ(ti) At]
135 v135 AL 3
+a, I35t ) ot + asxg (t,) 8¢ +ad(e,) at, (43)
135 - 135 — —_ 135 L
xg (t; + 48) xg (£) (1 —a, At —a @) At] + a,X135(t) st , (44)
135 “10 135
Xp3°(t, + 8t) = ———————— X!3%(t. + at) , (45)
¢ a.. +a, Gy ¢ °F
11 12 1
135 - 135
[Xg (t,z:)]eff WXg (ti) . (46)

In these equations, 1135 is the concentration of !35I in the circu-
lating salt, and X!3% is the concentration of 135%e, with subscripts s,
g, and b representing the components in solution, in the graphite pores
and in the circulating helium bubbles respectively. The parameters g

through a,,  were determined according to the analysis described in the

12
preceding pages and depend on the fission yields, radioactive decay con-
stants, mass—transfer coefficients, bubble characteristics, and external
stripping efficiencies. The factor ¥ is an importance-weighted shape~
correction factor for the distribution of !'3°%e in the graphite pores
mentioned above. At full power, this parameter was approximately equal

to 0.8.

The approximate model evolved in the manner outlined above (based on
an assumed nominal behavior of the circulating gas bubbles) proved ade-
quate to describe the 135%e reactivity behavior during operation with 235y
in the salt; however, as power operation of the MSRE progressed, evidence
was accumulated from special experiments designed to study the effects of
operating conditions on entrained gas behavior, and particularly from
operation with 233U in the salt, which indicated that the behavior of the

entrained gas was actually more complex than implied in the preceding
41

model. This latter evidence suggests that the gas bubbles circulating
under normal operating conditions with 235y may actually have averaged
only about one-third that indicated by the preceding analysis; furthermore,
it suggests that cover-gas solubility effects may have been significant
with helium cover gas at these low void fractions. Some of this evidence
will be described later in this report in connection with the discussion
of the reactivity balance data. Work is being completed on a refined and
extended model to describe the behavior of entrained gas and 135%e in the
MSRE. 18

In addition to its indirect influence on the reactivity through re~
duction of the 139Xe poisoning, the entrainment of undissolved helium in
the circulating salt also directly affects reactivity by increasing the
neutron leakage from the reactor core, This "fuel-salt density coefficient
of reactivity" was calculated as part of the analysis of core physics char-
acteristics. The value obtained was —0.23% 6k/k for 1 vol % circulating
gas bubbles. For a circulating bubble fraction between 0.1 and 0.15 vol %,
this would correspond to a reactivity effect in the range —0.023 to
—0.035% &k/k. T1If the actual gas civculating through the core were less,
as indicated above, this reactivity effect may have averaged only about
one-~third this amount during much of the reactor operation.

Recause the actual amount of gas in circulation appeared to vary
somewhat during operation, depending on conditions such as the liquid
level in the fuel-pump bowl, the transfer rate of salt to the overflow
tank, and the temperature,le’19 the absolute magnitude of this reactivity
effect was never well enough established to be explicitly included in the
on-line reactivity balance calculations. The effect is therefore included
in the residual reactivity term in the experimental results presented in

later sections.

Other nuclide transmutation effects

The seventh term on the right-hand side of Eq. (36) defines a cate-
gory that includes the effects of changes in the concentrations of 6Li,
23”3236’238U, 239,240py, and nonsaturating fission product peisons, all
in the fuel salt, and residual 0B in the graphite. (The term "nonsatu-

rating' is applied to describe this fiscion product group, since neutron
42

absorption cross sections of the first relatively stable member of these
fission product chains are sufficiently small and the MSRE flux levels
sufficiently low that burnout and production of secondary products were
negligible during MSRE operation,) The similarity of the reactivity
changes associated with this category is in the dependence on the integral
flux-time, or exposure, rather than the details of the power level vari-
ation during a single run. This net reactivity effect must be calculated
from basic theoretical considerations. The purpose of this section is to
provide a summary of the formulas used in calculating the changes in iso-
topic concentrations and in converting these changes to reactivity effects.
As a by-product, the assumptions in the computational model will be
exhibited,

The differential equations governing the changes in the isotopic

concentrations of the salt constituents are as follows:

 

 

 

 

dN6 R
3 = W, for OLi, 47)
v, ., -
dé = "quoquS for 23%y, (48)
dN26 .
dt ¥6%06% T oyl ,s for U, (49)
dll , n
d£ = MNZBUZSQS for 238U, (50)
s 5 » 239p,, *
Gi7 T uoluely t NpgT,9%, for Pu, (51)
%Tig 5 5 - 240p -
— = QN@0040®3 + ng(oqg'“ qu9)®8 for Pu. (52)

 

*
Corrections due to transients and neutron losses in 239Np were small

enough in the MSRE to be neglected.
43

In the preceding equations, X represents the nuclide concentration in
the salt, @S corresponds to the neutron flux averaged over the euntire cir-

culating fuel volume, represents the 235y fission density in the salt

FZS
averaged over the fuel volume, and o, is the ratio of productions of 236y
to fissions of 23%U., The reaction cross sections, 3, are averaged over the
reactor neutron spectrum (subscript f refers to the component due to fis-
sion); their precise definitions and a description of the method of calcu-
lation are deferred te the following section.

The buildup of the nonsaturating group of fission product poisons was
represented by the approximate equation (53). 1In this equation, the pro-
duction rate of the 72th nuclide is proportional to the product of the fis-
sion rate in 23°0 and the yield from fission.* The removal rate due to
neutron capture is equated to zero. Thus the differential equation de-
scribing buildup of the fission product inventories, ignoring effects of

external removal of these products from the circulating fuel system, was

7
7
Vg o,

@7 ¥ (53)
Actually, MSRE experience indicated that certain of these fission products
were continuously removed during operation, and corrections needed to be

introduced into inventories calculated according to Eq. (53) to account
for these effects. HNumerical results obtained in these calculations are
described later in this section.

In contrast to the nuclide constituents of the fuel salt, residual
108 in the graphite was exposed to the local neutron flux ®g(r,t), so the
burnout rate was position dependent throughout the core. If Nlo is the
concentration of 198 in the graphite, the appropriate differential equa-

tion 1is

 

at. 1010

-~V T @g(r,t) . (54)

 

*Corrections corresponding to the small fission product contribution
from fission of 23%Pu were neglected,
44

The flux level and the total exposure of the 235U fuel charge
(v70,000 MWhr) were small enough that the wmagnitude and energy spectrum
of the flux corresponding to a given fission rate (or power level) was
very nearly independent of exposure. Consequently, the spectrum-averaged
reaction cross sections could be reasonably assumed to be invariant during
operation. However, the flux in the preceding equations was actually time
dependent, since the power level varied during reactor operation. This

is commonly accounted for by making the change of variable £ - u, where

T ; ;
u = [ o (') dt . (55)

The relation between the total number of fissions during an operating

interval and the time integral of the flux during that interval is

 

 

N A 5 A n
0 Uf25 -025u N28628 Ofug
Total fissions = V N — (1 — e Yy + ¥V N0
525 5 \" o5, -5 5
25 49 28 L9
=0, gt “4g qug ~C, gl
x (1—e ) + V) |- - — (1~ e Y, (56)

ug — %28/ Tuq

where NO refers to the concentrations at the beginning of the interval and
VS is the total volume of salt in circulation. Because the total fuel
exposure was sufficiently low, the time-integrated flux is approximately
proportional to the total fissions; that is, to the time integral of the

power. Hence the additional simplification

u = o7 (57)

could be used, where ¢ is the volume-averaged flux mnormalized to unit
power level and 7 is the time-integrated power. Moreover, the contribu-
tion from 23%pu fissions was quite small (increasing from zero to less
than 3%, maximum) during the entire period of power operation with 235U,
so the average fission rate in 235y per unit power was essentially con-
stant. With the above assumptions, the integration of the preceding dif-
ferential equations is a straightforward exercise, and the resulting

formulas are those listed below:
 

 

—06¢T
AN, = “Ng (1 —e ) for °Li .
0 21T 23t
- ‘ — +
AIVZI+ —qu (1 e ) for U
. ~826¢T
~0,. 0T  a._ Foc (L —e )
26 25 25
M, = N (1-—e ) = for 236y
6 26 ~
O6?
"8?8¢T
= 70 - 2 238
ANZS N28 (1 e ) for U
-G, 4T N & 5. 4T -0, 6T
L 28 2 4y 5
AV = -0 (1 —¢ 2 ) + 28 (e 8 ? ) for 23%puy,
49 49 5 -
49 28
. 0,4 — 0. A A
AV, = -0 (1 -~ —OHD¢T) + 0 P T e ”U”9¢1)
40 0 ¢ n9 a4 ’ ¢
be ~ Ju0
wo Ong 3 — 0 -G ¢ -0, 07
28 b 2
- - 9 ;fug (o o' 8 )
“49 T 958 |T28 7 i
5 - . .
g fug -0, 4T ~3 T
- ¢ 0 —2 "y for 240pu ,
Shg T Tyq
7 — .
ANF.P. = YiEZST for non-saturating F.P.,

M= —N?O (1 ——e”glo¢g(r) 5y for 10g .

(58)

(59)

(60)

(61)

(62)

(64)

(65)
46

In Egqs. (60) and (64) the quantity F25 is the volume-average fission
density in the salt per unit power, and ¢g(r) is the local neutron flux
in the graphite, also normalized to unit power level.

The theoretical considerations required to convert the concentration
changes to reactivity effects were described in Section 2.3, where for-
mulas were developed in terms of the two-group diffusion model.* For a
neutron absorbing material distributed in the salt, the reactivity effect
was given by Eq. (27). Similarly, for a change in the concentration of
a fissile material, the reactivity effect was expressed by Eq. (31h).

The single exception included in this nuclide category, which re~-
quires special treatment, is that of the 108 burnout in the graphite. In
this case, the spatial distribution of burnout is of significance. If
SN represents the local change in 10g concentration, the reactivity

10
equation is (still using the two-—group notation)

- IVm ¥ CSNlo(l - Fs)0a1¢1 dv - fvm v, 6N10(1 - Fs)gazq)z av
Aplo = , (66)
Jy pvip oy +oyvig e ) v
where Vfi represents a volume enclosing only the moderated (graphite
channeled) region.

To better exhibit the spatial effects in this calculation, it is
convenient to separate Eq. (66) into the product of a shape-factor cor-
rection and the reactivity change that would be associated with uniform
burnout of the boron at a rate determined by the spatial-average flux in

the graphite., Defining Sfigo as the change in concentration if the burn-—

out is uniform, we may rewrite Eq. (66) as follows:

GNlo (Jmlgal +'Jm20a2)
Ap = —5(1) s (67)

Lo Nf (jlvafl + vof2>

 

 

*The actual neutronics calculations were based on a four-group dif-
fusion model, as described in Chap. 4. This generalization of the re-
quired reactivity formulas is straightforward; hence in this section we
continue the use of the two-group formulas to take advantage of the
abbreviation thereof.
47

where
IVm b, (L= F )¢, AV
J = (68)

[, v F ¢ dv
VR 1 82

 

and
[y v, —F)4, dv
joo o= = ; (69)
[, V. F ¢, dV
VR 1 &8°2

 

jl is defined by Eq. (29), and the shape factor is defined by

fvfiwl 8N, 018, av + thwz S Aav

ST = . (70)

e

o (;V# b 9,0 AV F IVfi Y, 942% dv)

 

In Eq. (70), GNlo is calculated according to Eq. (65), and the same equation
defines 6N10, with the spatial flux distribution replaced by the volume-
average flux in the graphite. Expressed in this product form, Eq. (67) is
also more suitable for approximation of the spatial flux distribution and
calculation of S(I) by numerical integration procedures, Results of nu-
merical calculations of this spatial burnout factor are shown in Fig. 10.

In order to apply the theoretical formulas of this section, infor-
mation is required concerning initial concentrations of the various nu-
clides, their reaction cross sections averaged over the MSRE spectrum, and
the flux-adjoint flux-~product integrals entering into the reactivity form~-
ulas. Brief descriptions of the sources of this information are the topic
of Chapter 4.

The results of calculations of the individual reactivity effects

%
lumped in this category are shown in Fig. 11. These numerical values

 

N

hThe on-~line calculations of reactivity effects described in Sect. 5.2
were actually based on an earlier version of Fig. 11. The numerical values
in this figure represent current best estimates of the actual changes during
operation with 2357, They reflect cross-section data improvements made
gince the termination of operation, modifications further discussed in
Sect, 5.3 in comnection with refinements in evaluations of long~term re-~
activity trends,
48

ORNL-DWG 71-4245

 

 

 

 

 

 

 

 

 

 

 

 

18 ‘[
|

17 \\ ,,,,,,
T
2 g \ e _
2 AN
X \
=z
g 1.5 p—— ~ -7 —
W \
f
44 ‘ S
o \
o \\
13 b= . i
5 T~
@ \
T o
k2!

 

 

 

 

 

 

 

 

 

 

 

0 10 20 30 40 50 &0 70 80

TIME-INTEGRATED POWER (102 MWhr)

Fig. 10. Spatial correction factor S(7) for '°B burnout reactivity
effect in MSRE graphite.

ORNL-DWG 71-49472

FPLUTONIUM-23

=
™~
o
[78)
2
)
O
=
<
T
o
o~ URANIUM-236
}..
=
!...
=
1 NON-SATURATING

FISSION PRODUCTS

 

o 10 20 30 40 50 60 70 80

TIME—INTEGRATED POWER (1000 MWhr)

Fig. 11. Reactivity changes due to long-term nuclide changes in
MSRE during operation with 23°U; no fission product removal.
49

are also normalized to the 239U loading attained at the end of the zero~
power experiments and are similar to the normalizations of other reactivity

effects shown in Figs. 4 and 7. The curve labeled "

net'' is the algebraic
sum of these effects and would represent the contribution from this cate~
gory within the overall reactivity balance., As was mentioned ahove, how-
ever, one additional effect needs to be considered; namely, the external
removal of certain fission products from the MSRE fuel salt. The calcu-
lations in Fig. 11 were based on the assumption that all fission products
and their daughters remained in the medium in which they were produced.
Many of these fission product chains, however, include noble gases (xenon
and krypton) with substantial half~lives. Since these gases are largely
insoluble in the fluoride salt melt, they tended to escape into the off-
gas system. In addition, there is evidence that significant fractions of
some noble metals (notably Mo, Te, Ru, Te, and Nb) were also removed by the
off-gas system, either as volatile fluorides or as colloidal metal parti-

20

cles. Another mechanism for loss of poisoning, which occurred to some

extent, was the plateout of fission products in surfaces outside the core

2l A factor that enhanced the poisoning effect of some fission pro-

region.
ducts was the diffusion of some volatile species into the pores of the
graphite. It is clear from the above that a quite detailed model of fis-
gion product behavior would be required for a precise description of fis-
sion product poisoning effects in the MSRE. Although wuch information was
gained from analysis of MSRE operation, such a model is not cuxrrently
available.

To obtain some indication of the effect of fission product loss on the
net poiseoning, we made calculations for two idealized cases. 1In one case,
we assumed removal of 1007 of the noble gases and in the other, concurrent
removal of both the noble gases and the noble metals. (The only members
of the latter category that are significant neutron poisons are molybdenum,
ruthenium, and tellurium.) Figure 12 shows the changes produced in the
net reactivity curve of Fig. 11 when these amounts of external removal of
fission products are considered. The largest change resulted from removal
of the noble gases. (Note that the direct poisoning by 135%¢ 15 treated
separately in the reactivity balance and is therefore not included in this

analysis.)
50

ORNL-DOWG 71-4343
—-—-REMOVAL COF Xe, Kr, Mc, Ru, Te

REMCVAL OF Xe, Kr

FISSION PRODUCT REMOVAL

REACTIVITY CHANGE {% Bk/k}

 

0 10 20 30 40 50 60 70 80
TIME-INTEGRATED POWER (1000 MWhr)

Fig. 12. Effect of fission product removal on net reactivity change
induced by long-term nuclide changes.

In the absence of detailed quantitative information on the behavior
of all the fission products, we assumed for the reactivity balance calcu-
lations removal of all the noble gases and none of the noble metals. Thus
the middle curve in Fig. 12 was chosen to represent the net reactivity ef-
fect of this category of long-term core changes.

For the on-line calculations, an approximate formula was used to ob-
tain the net reactivity effect of this group of changes. This was of the
form

bo = A+ AT+A e V' va e %, (71)

Numerical values of the parameters in this formula were determined accord-
ing to the analysis described in this section; renormalization adjustments
were also made each time the fuel was drained and mixed with the residue
of salt in the fuel drain tanks, which had not been exposed to the flux

during that run.

3.3 1Influence of Graphite Irradiation Damage
on the MSRE Reactivity Balance

 

One effect that was expected to occur but was not specifically in~-
cluded in the reactivity balance calculations was associated with dimen-
sional changes in the core graphite due to fast-neutron irradiation. The
type of graphite used in the MSRE undergoes the changes indicated in

Fig. 13 (Ref. 22). The result is that the distribution and the amount of
51

ORNL- DWG 69-10228

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2 -
O L
-0.2
2
., —0.4
D
=
T -0
5 .
1
I o8
g o2
=
W
5 o A
g 1/{? -
-0.2 l— %@fi .
WM%Z%z?zz?»
ol : g
-0.6 .
0 | 2 3 (x10%"

NEUTRON FLUENGE [neutrons/cm?® (£ > 50 keV)]

Fig. 13. [Irradiation-induced dimensional changes in MSRE grade CGB
graphite.

fuel salt in the core can change both from graphite volume changes and
from the bowing and displacement of the graphite stringers. The dimensional
changes can potentially affect the core structure in three ways: (1) a
direct change in the linear graphite dimensions that is dependent on the
neutron fluence, (2) various degrees of bowing of individual stringers,
depending on the flux gradient across the stringer and on the total flu-
ence, and (3) displacement of the lower end of each stringer by the di-
mensional changes of the lower horizontal bars. Since the net changes in
the core structure as the hundreds of graphite bars bow in various modes
due to the nonuniform flux distribution are quite complex, a computer pro~
gram was developed in an attempt to describe this situation in detail.
Relative changes in the volume fraction of salt and graphite at each core
location were obtained, and the associated reactivity effects were esti-
mated as weighted averages of these changes over the core volume. The

essential features and results of the calculation are outlined below.
52

Calculation of graphite displacement

The individual graphite stringers in the core are bowed by changes in
the differential expansion (or contraction) that occur through the thick-
ness of the stringer as a result of the flux gradient, If the stringers
were unrestrained and the neutron flux and graphite distortion curves were
linear across the thickness of the stringer, there would be no stress, and
the deflection at any point could be calculated by conventional beam-
deflection equations. We will consider the geometrical relationships indi-

cated in Fig. 14, where the radius of curvature is

 

C
R = m 5 (72)
and
1 d2y/dm2
3/2
[1 + (dy/dz)?]
If dy/dx is small, (dy/dz)? is negligible, and
2 » .1 _ AL/L
dcy[dz= = > oo (74)
The deflection is
y = R G g (75)

ORNL-DWG 71-4948

GRAPHITE
STRINGER

 

 

-~

Fig. 14. Stringer deflection geometry.
53

and since there are dimensional changes at both positions 1 and 2,

(AL/L)  — (AL/L)
y = [f 2 Llde® de . (76)
c

 

The change in differential expansion across a stringer, and therefore
the deflection of the stringer, depends on the slope of the radial flux
profile and on the slope of the axial (parallel to extrusion) distortion
curve for the graphite shown in Fig. 14. TIf there were no contact with
adjacent stringers and no local flux perturbations, any stringer in the
core would deflect in a radial plane from the reactor center line.

A computer program was written to calculate the components of the
radial deflection at 2~in. intervals along the length of each stringer and
to add these to the respective displacements of the lower lattice bars.
The graphite distortion data and the radial flux profile were fitted to
cubic equations for use in the computer program. Equations were obtained
for both the upper and lower boundaries of the graphite distortion curves,
The axial flux distribution was represented by a chopped cosine curve.

When the lower axial damage curve was used, the deflections were out-
ward for all the stringers, and there were no interferences indicated un-
til an integrated power of 150,000 MWhr was reached. However, when the
upper curve was used, the outer stringers with lower neutron exposure and
with a positive slope on the distortion curve deflected inward against the
neighboring stringers. In this case the outermost stringers were allowed
to retain their calculated deflections, and adjustments were made pro-
gressively inward on the neighboring stringers to eliminate any interfer-
ence. No attempt was made to calculate the elastic deflections that would
occur when two or more stringers were in contact with each other. However,
the above method of adjusting the deflections was found to be unrealistic
above 45,000 MWhr and caused solid packing of the graphite over a greater
fraction of the core than could actually occur. We did not attempt fur-
ther refinements in the mathematical model to correct for this, since
these calculations were to be used only as rough estimates of the possible

magnitudes of reactivity effects introduced by graphite irradiation damage.
54

After the deflections were calculated and adjusted as above, the
changes in graphite and salt volumes were calculated at each core location,
The graphite and salt densities based on core volume were then calculated
at each location from the volume changes. The relative changes in salt
and graphite density at each core location were output on punched cards

for use in calculating the reactivity effect,

Conversion to reactivity effects

The reactivity effects of the relative density changes described above
were determined by weighting these changes according to the nuclear impor-
tance of the core location and integrating over the core volume. This cal-
culation was similar in form to that for the !YB burnout reactivity effect
described in Section 3.2, except that additional terms were required to
account for the effects of these density changes on the moderating proper-—
ties of the core. The total reactivity change was calculated from the

following equations (in two-group theory):

 

1,2 s L (ST

e = oo |9 [-2) + 8 [-8 , (77)
» - ’LJ N 4 » /LJ N LN
Tad g ’1g g /g

where the importance-averaged density changes in the graphite (and simi-

larly for the salt) were

 

SN
T IV wj(fifi) b, dv
(_N—fi) -2 7 . (78)
g iJ
fvg bio, AV

In these equations N is the relative density, the Kij are veactivity coef-
ficients obtained from neutronics perturbation calculations, and s and g
refer to salt and graphite respectively. A second computer program was
written to perform the integrations and to calculate the total reactivity
effect, Imn this program, the flux profiles were input as tabular data

and interpolated as necessary. The values of 8N/N for salt and graphite

were read from the card output of the graphite distortion program.
55

Results of calculations

 

The reactivity changes predicted by this model were calculated at
various exposures up to 200,000 MWhr, well beyond the actual exposure
during MSRE operation with 235y, These results are shown in Fig. 15.
Calculations with either the upper or lower limits of graphite damage data
indicated a long-term positive trend in reactivity. The initial decrease
in the curves obtained with the upper graphite damage limit is caused by
the positive slope of the graphite damage curve at low neutron exposures,
This indicates that the stringers would tend to deflect inward at low
values of integrated power and squeeze fuel out of the core; then, as the
integrated power increased, the central stringers would begin to deflect
outward, bring fuel back into the central region of the core, and increase
the reactivity. For the upper-limit damage curves, there is increasing
uncertainty in the slope of the reactivity change beyond'about 45,000 MWhr
because of the problem of accurately assessing interference effects between
the stringer deflections.

In addition to the reactivity effects, the computer program also out-
put the various dimensional changes within the core graphite and the magni-
tude of any interferences. The upper graphite damage curves produced in-
terferences at all integrated powers, while the lower curves produced
interferences at 150,000 MWhr or above, At an integrated power of 100,000

MWhr, the maximum bowing was 0.036 in., and the overall length change

ORNL - DWG 69- 10227

 

 

 

0.5 T‘fihv """"""" W """" ( """ - ]’“%‘
= 0.4 s L CALCULATED USING LOWER -t s
< ' LIMIT GRAPHITE DAMAGE cuwvgs-hfli,/”
2 03 b ol
> ! : 1
= CALCULATED USING UPPER Lt fiz,z“
Z 0.2 b UMIT GRAPHITE DAMAGE CURVES ==< ‘*w::;; -------
> ; \>,
§ 0.1 Lfi’/_,e"_/‘ ‘,1‘.....5...‘.;“‘ ,,,,,,,
& _.—/T,. // ‘
o —
T ’-—‘/
--——'I—"
-OJ el el L

 

 

 

 

 

 

 

 

 

 

 

C 20 40 60 80 100 120 140 160 180 (xi0%)
INTEGRATED POWER (Mwhr)

Fig. 15. Reactivity effect of irradiation-induced dimensional changes
in MSRE graphite.
56

varied from ~0.076 in, at the center line to ~0.006 in. at the periphery

of the core. The maximum interference between stringers calculated from
the top graphite distortion curves was 0.049 in. calculated at 150,000 MWhr.
If all this interference were relieved by the deflection of one stringer,
the calculated bending stress in that stringer would be 920 psi.

Because of thée obvious difficulties involved in modeling the actual
core behavior under prolonged graphite irradiation, the calculations de-
scribed should be viewed as attempts to bracket the possible reactivity
component associated with this effect. These results will be further dis-
cussed in connection with the interpretation of long-term reactivity

trends (Sect. 5.3).
57
4, SOURCES OF DATA AND PARAMETER CALCULATIONS
4,1 1Initial Nuclide Concentrations

The basis for calculating initial nuclide concentrations for this
problem was the fuel-salt "book" inventory and composition records at-
tained at the beginning of the power runs. As reported in Ref. 23, this
was LiF-BeF ~ZrF,-UF, (64.88-29.26-5,04-0,82 mole %), in which the uranium
contained 33.241 wt % 235U, The fuel-salt density required to convert the
chemical composition to nuclide concentrations was 145 1b/ft3 at 1200°F
(based on evidence discussed in Ref. 11).

The source of data for determining the b1.1 isotopic content in the
initial fuel salt was a set of mass-spectrographic analyses of the LiOH
batches from which the LiF for the fuel salt was prepared. These assays
gave an average of 0.0074 at. % SLi.2"

Data regarding the concentration of residual boron in the MSRE core
graphite was taken from Ref. 25. As reported there, the chemical purity
analysis of grade CGB graphite was 9 x 107° wt % boron (natural isotopic
abundance, 19.8% 10B). The average density of the MSRE graphite at 1200°F
was assumed to be 1,86 g/c:m3 in all calculations.

The source of data for the 23%U and 235U concentrations was Ref. 26.
The uranium used to prepare the enriching fuel concentrate contained an
average of 0.98 wt % 23%U and 0.38 wt % 235U, These were sufficiently
small that the reactivity balance calculations for the 235y loading were

quite insensitive to the initial concentrations of 234y and 236y,

4.2 Average Reaction Cross Sections

For those nuclides uniformly distributed in the fuel salt, a general
expression for the reaction rate in the Zth nuclide was given as Eq. (9)
in Section 2.2. This characteristic of uniform distribution is basic to
these circulating fuel systems and applies to all the important constitu-~
ent nuclides in the salt. For our calculations of the time~dependent
changes in these nuclides, we have found it convenient to define a micro-

scopic reactor-—average cross section by rewriting Egq. (9) as

A

R, = Nioi%hVS 5 (79)
58

where we have made the definitions

o0

fi. dv-/; dE 0. (B) (x,F,t) F(x)

 

 

5.8 = I 7 (80)
f dvf 4r ¢(c,E,t) F(r)
Ve o
and
E
f dvf ¢ dE ¢(r,F,t) F(r)
v 0
R
- . ) (81)
5 (£
2, () .

In these definitions, Ec is a convenient cutoff energy chosen high enough
to effectively separate the slowing—down energy range from the thermali-
zation range, @th is the average thermal flux to which the salt is exposed,
and Vé is the total volume of salt in circulation. [Note that a similar
definition could, if desired, be made on the basis of the total neutron
flux at all energies, ¢, simply by replacing Ec by « and ch by ¢ in
Egs. (79) through (81).]

As defined by Eq. (80), the average reaction cross sections depend
on time through the relative changes in the neutron flux spectrum; however,
in the MSRE the variations in salt composition during power operation were
sufficiently small, and the flux perturbation due to control rod movements
sufficiently localized, that the changes induced in the average cross sec-
tions were quite small (see Tables 1 and 2 and related discussion below),
Hence, close approximation to the values of G could be obtained by calcu~
lating the spectrum for average operating conditions during the power runs.

To apply the preceding definitions to reaction rate calculations, it
is necessary to approximate the neutron spectrum ¢(r,£F). In our work with
the MSRE, we used the GAM-II and THERMOS programs to produce four broad-
energy—-group average nuclide cross sections, where the averaging is with
respect to an approximate core spectrum calculated by the programs.27’28
We then applied the latter cross sections in few-group diffusion calcula-

tions with the EXTERMINATOR program29 to approximate the variation in the
59

flux spectra in the peripheral, salt-containing regions of the core (i.e.,
the upper and lower plenums and the radial downcomer). Here, a two-
dimensional (#,Z geometry) representation of the MSRE core was used, with
four broad energy groups (one group spanning most of the slowing-down
range, two epithermal groups, and one thermal group). The geometric model
is shown schematically in Fig. 16.

Approximate expressions corresponding to the defining Eqs. (80) and
(81) can readily be cobtained to describe the procedure of calculating

average reaction cross sections, They are

-n_K
;Z VF 9%,
. Vi
~ e (82)

and

where Vk and Fk are the volume and salt volume fraction for thekkth sub-
region over which the volume fraction of salt is coastant and ¢ 1is the
EXTERMINATOR flux in broad group n averaged over subregion % (@Eh is the
average thermal flux in region k). The broad-group nuclide cross sections

are defined by:

 

n+l
dE ¢a(b) Ui(E)
07: = ] s (84)
Zi?’H-lT.
/ ax ¢ (@)
B
¥

where ¢G(E) is the approximate neutron spectrum in the graphite-moderated
core as obtained from multigroup calculations with the GAM-TII and THERMOS
programs. The integrals in Eq. (84) are also calculated in terms of fi-

nite sums, with up to 99 energy groups used to span the slowing-down range
60

ORNL-DWG 71--9807

 

 

 

N

N in\\\\
©
ny

4 219.20

 

 

 

 

 

 

 

 

 

 

 

 

COMPOSITION REPRESENTS DIMENSIONS (R x 2)
T T - fem}
t GRAPHITE~SALT LATTICE 70.49 x 166.45
2 TOP PLENUM 73.66 x 23.85
3 BOTTOM PLENUM AND STRUCTURE 73.66 x 2327
4 RADIAL DOWNCOMER 2.54 x 166.45
5 CORE CAN 0.64 x 16645
6 REACTOR VESSEL (TOP AND BOTTOM) 2.85%
REACTOR VESSEL (SIDE) 1.43%
7 CONTROL THIMBLES (PQISON WITHDRAWN) 1.80 x 147.25
8 CONTROL ELEMENTS (POISON INSERTED) 1.80 x 29.08

*LENGTH DIMENSIONS ON FIGURE

Fig. 16. Geometric representation of MSRE core used in four-group
diffusion calculations. Poison elements withdrawn to upper limits; not
to scale.
61

and 30 to span the thermal range. The cutoff energy, Ec’ separating these
ranges was chosen to be 0.876 eV for MSRE calculations.

In Table 1, are listed the average cross sections calculated by these
methods. Two sets of values are given, with the left column representing
the averages obtained from cross-section libraries in usage early in the
course of MSRE operations and the right column representing averages from
more recent data libraries (ENDF/B) currently in use for molten-salt reac-
tor studies. A discussion of the significance of these and other cross-
section comparisong in interpreting nuclear operations data for the 235y
loading is given in Ref. 30. Most of the on-~line calculations described
in the next section were based on the older data set given in Table 1;
however, a recent reevaluation of the long~term reactivity trends for the
2350 loading was made from values of & based on the newer data, and these
results are described in Section 5.3.

The calculations listed in Table 1 were made for the reactor spectrum
corresponding to the minimum critical loading conditions (core isothermal
at 1200°F and all control rods fully withdrawn). These average cross

sections should be corrected for changes caused by increasing the uranium

Table 1. Calculated average reaction cross sections in
the MSRE thermal flux with 23°U at minimum
critical loading conditions

 

Average cross section (b)

 

 

Nuclides Pre-1965 data 1969 calculations
°Li 461.02 L66.42
10g 1870.35 1881.13
135%e 1.18 x 10% 1.18 x 106
149gy 3,58 x 10" 3,58 x 10"
151gy 2.63 x 103 2.63 x 103
234y 138,19 130,40
2357 (abs) 340.53 333,88
2357 () 270.15 267.88
236y 52.76 51.11
238y 26.32 27.67
23%py (abs) 1470.1 1456.1
23%y (f) 874.5 901.1

240py 1267.9 1287.6

 
62

concentration to reactor operating conditions and also the insertions of
the control rod to compensate for the excess reactivity. These changes
increase the average epithermal-to~thermal neutron flux ratio in the re-
actor and thereby increase the reactor—-average cross sections [since they
are defined by Eq. (80) on the basis of the thermal component of the flux].
In Table 2 we have indicated the relative increases in calculated average
cross sections of the most important nuclides corresponding to the maxi-
mum 2350 concentration and rod poisoning experienced in the 235y run.

This occurred at the end of the zero-power experiments and before the fuel
was drained, when the 235y concentration had been increased by nearly 10%
and the regulating rod was near full insertion. Although further uranium
additions were made during the 235y run, at no time during operations were
these conditions exceeded. 1In accord with the relative increase in epi-
thermal flux, the maximum changes in Table 2 may be seen to be associated
with those nuclides with strong resonance-capture components.

During the actual course of reactor operation, the uranium concentra-
tion and the rod insertion varied in accordance with the fuel burnup and
additions and with the various reactivity changes in the core. The changes
listed in Table 2 tend to bracket the actual variation in 3, and, for pur-—

poses of calculating the reaction rates, the actual variations can be

Table 2, Relative changes in average reaction
cross sections for a 10% increase in #3°U
concentration, with compensating control
rod insertion

 

 

Nuclide Cross—section ratioa
234y - 1.063
233U (abs) 1.018
235y (£) 1.014
236y 1.089
238y 1.094
23%9py (abs) 1.004
23%py (£) 1,004

 

a . . . .
Cross section at maximum uranium loading con-
ditions divided by cross section at minimum critical
loading.
63

closely approximated either by linearly interpolating the results in
Table 2 on the basis of the #35U concentration or by assuming a fixed
average composition during operation.

The source of data for cross sections of the nonsaturating group of
fission products was Ref. 31.* This provided a summary of fission yields,
cross sections at 2200 m/sec, and resonance-absorption integrals for these
nuclides, These data are listed in Table 3. The group includes all fis-
sion products (exclusive of 135Xe, 1495m’ and 1515m) known to have non-
negligible cross sections., With the single exception of 1”7Pm, only the
direct products from fission (or daughter products of precursors with
very short half~livesg) are listed in Table 3. As already indicated, it
is reasonable to neglect the secondary products because we were dealing
with relatively small fuel exposures. The final nuclide entries in

Table 3 are "pseudonuclides,”

representing groups of fission products in
which saturation effects are either negligible (NSFP) or quite small for
long fuel exposures, as well as for short exposures (SSFP). Explicit tabu-
lations of the nuclides included in these groups are given in Ref. 31.

For the purposes of this work, we neglected any non-1/v behavior of
the absorption cross sections over the thermal energy range exhibited by
the fission product nuclides of Table 3. These are generally small cor-
rections that tend to be unimportant compared with uncertainties associ-
ated with transport and removal of fission products in the MSRE fuel sys-
tem. (Influence of these latter effects is discussed in the next section.)
Thus, averaged cross sections for the thermal group were obtained by multi-
plyving the 2200-m/sec cross section by the thermal-spectrum average for a
1/2 nuclide that had unit cross section at 2200 m/sec. This latter quan-
tity was calculated to be 0.435 in the MSRE spectrum. The average epi-
thermal cross sections were cbtained by dividing the nuclide resonance
integral in Tahle 3 by the lethargy width for the epithermal range.

As an approximate indicator of the significance of saturation effects

in calculating the fission product concentrations, one may estimate the

 

%
Reference 31 is largely based on the more extensive data compilation

in ref. 32, with emphasis given to useful simplifving approximations for
reactor physics calculations.
64

Table 3. Cross section and yields of significant
fission product nuclides

 

 

Nuclide vield o (2200) REEEZZ?ST Half-life
(%) “ above 0.414 eV

82Ky 0.28 45.0 196.0 %

83Ky 0.544 205.0 189,2 %

103gn 3.0 149.0 1059.0 w0

105pn 0.90 15,000 7261.0 36 h

109pg 0.03 87.0 1447.0 %

131xe 2.93 120.0 811.4 0

133%e 6.59 180.0 92.33 5.27 d

143Na 6.03 324.0 134.0 o

145Nd 3.98 60.0 314.6 o

I 7pm 2.36 235.0 2279.0 2.65 y

15251 0.281 208.0 2242.,0 o

15354 0,169 450.0 432.1 16 v

155gy 0.033 14,000 6787.0 1.7 y

NSFP 126.0 1.111 7.354

SSFP 29.8 13.986 76.744

 

aDecay product of 11,1-d 147N4q.

cross section required for a 10%Z reduction in the otherwise linear rise
in concentration corresponding to zero absorption. For a maximum fuel
exposure equivalent to 70,000 MWhr at MSRE flux levels (somewhat larger
than the exposure of the 235U fuel loading), the absorption cross section
for this reduction would be about 1000 b, Thus, of the nuclides listed
in Table 3, only 105h and !5%Eu would be expected to exhibit significant
saturation effects. Due to its short half-life and high cross section,
105ph would quickly saturate, with an associated reactivity effect of
about —0,004% &8k/k. Since primary interest was in effects of larger mag-
nitude, this nuclide could be safely excluded from the fission product

poisoning concentration. Europium-155, on the other hand, did mot reach
65

its saturation concentration, but neutron absorption would have reduced
the otherwise linearly rising concentration by about 407 by the end of the
235y fuel exposure. This gave a total reactivity effect of about —0,014%
Sk/k by the end of the exposure period, which was still quite small.
Rather than treat this as a special case, we approximated this correction
by including the nuclide with the remainder of the group of nonsaturating

fission products and reducing its effective fission yield by 25%.

4,3 Neutron Flux Normalization

 

The numerical evaluation of the average neutron flux level is basic
to the theoretical calculations of nuclide changes in the reactor. The
principal determinants of this quantity are (1) the mean energy recovered
per fission, assumed to be 200 MeV;33 (2) the volume of salt in circula-
tion, 70.5 ft3; and (3) the inventories of 235U and 23%Pu in the loop at
operating conditions. The average thermal flux per megawatt for the cir-

culating system is then defined by

A
¢ = . (85)
th A N
(N25 0f25 + Nug Ufug) VS

 

For the mean fission energy given above, the conversion factor 4 is equal
to 3.12 x 101® fissions/MWsec. From this calculation, we obtained ¢th =

6.4 x 10} neutrons cm™2 sec—!

per megawatt, near the start of power opera-
tions with 23%U., The corresponding value of the power-normalized thermal
flux averaged only over the graphite was 2.0 x 1012, This flux was used

in the calculation of the 19B burnout effect,.

4.4 Flux—Adjoint Flux Integrals and Reactivity Parameters

 

The direct and adjoint flux distribution and their volume integrals
over various subregions of the reactor core required for calculation of
reactivity effects are standard output of the EXTERMINATOR program used
for the neutronics calculations. 1In accordance with the discussion in
Section 3.2, the volume integrals over regions of the core with fixed salt
fractions, together with the broad~group cross sections for the nuclides,

provide information sufficient to convert all nuclide changes to reactivity
66

effects., (For several nuclides, the reactivity effects of unit changes
in concentrations were directly output from the EXTERMINATOR program.)

For the numerical computation of the shape factor, S(I), in the re-
activity effect of boron burnout [Eq. (70)], we employed an additional
simplifying approximation. The neutron flux and adjoiont distribution in
the graphite region were represented by a fundamental mode approximation
with a "chopped" Bessel function variation in the radial direction and a
"chopped" sinusoidal variation in the axial direction. S(7) was then cal-
culated by numerical integration with an IBM 7090 program. The largest
discrepancy between the approximate flux distribution and results of the
detailed diffusion model calculations occurred in the radial flux vari-
ation; however, the latter calculations indicated that the flux perturba-
tion caused by the control rods remained fairly well localized near the
center of the reactor, and over most of the reactor volume the Bessel func-
tion approximation would be valid. Thus, this approximation was judged to

be adequate for the calculation of 5(7).

4,5 Effective Delaved-Neutron Fractions

In accord with the discussion given in Section 2.4, numerical evalu~-
ation of the delayed-neutron effectiveness is required for the interpre-
tation of measured reactivity effects. An approximate formula for this

34 of this effective-

calculation was given as Eq. (35). E£arly estimates
ness factor, ¥, gave a value of 1.04; this value was used for all initial
interpretations of MSRE reactivity measurements and for the on-line cal-
culations with the 235U loading summarized in Section 5. More recently,
near the termination of power operation with 235U, we reexamined3® the
evaluation of ¥y with a more sophisticated calculation (not available at
the time of the early calculations) of the difference in average age of
prompt and delayed neutromns., The age to the thermal cutoff energy was ob-
tained through use of the GAM-II program. For a salt-graphite composition
equal to that of the channeled region of the core and a temperature of
1200°F, the age calculated for the prompt fission neutrons was 251.9 cm?,
The calculated age difference between prompt fission neutrons and delayed

neutrons was 62.3 cm®. Based on a geometric buckling corresponding to a
67

cylinder 29 x 78 in. (RxH), the value of ¥ calculated from Eq. (35) was
1.086., This implies, for example, that the total effective delayed-neutron
fraction for use in interpreting reactivity neasurement with 2357 was 0.717%,
compared with the absolute delay fraction of 0.65% (Ref. 8, p. 920). Hence,
it implies that the magnitfides of all measured reactivity effects should be
larger by a factor of about 1.05 than the values based on earlier calcula-
tions of y. 1In the results reported in Ref. 11, this would have the effect
of increasing the difference between the experimental and calculated worths
of the MSRE control rods but would tend to bring the measured temperature
and 235U concentration coefficients of reactivity into better agreement
with calculated values,

We did not attempt to vrevise the numerical values from the oun~line
calculations to account for this change but did include its effect in re-
cent reevaluations of the long~term reactivity trends during the 235y op-

erations (see the discussion in Chap. 5).
68
5. EXPERIENCE WITH THE REACTIVITY BALANCE

5.1 Brief History of Nuclear Operations with 235y

 

A summary of significant dates in the chronology of nuclear operations
with 235U is given here Lo provide perspective for those readers least fa-
miliar with the details of MSRE operation. The dates and events are listed
in Table 4. In connection with the interpretation of long-term trends in
reactivity associated with power operation, it should be noted that the
first sustained power operation really began with the 30-day run starting
on December 14, 1966 (run 10). A full account of the chronology of the

MSRE test program and operation is given in Ref. 36.

5.2 Results of On-Line Reactivity Balance Calculations

The on-line calculations during MSRE operation were made with a Bunker-
Ramo 340 digital computer connected directly with sensors in the MSRE sys-
tem. This was used for routine scanning of various input signals and for
comparisons with alarm limits, data storage operations, and several routine
data-reduction operations, such as the heat balance and reactivity balance
calculations. The characteristics and functions of and experience with the

logger—computer may be found in Ref. 37.

Table 4. Significant dates and events in nuclear
operation of the MSRE with 2357

 

 

Event Date
First criticality June 1, 1965
End of zero-power experiment program July 3, 1965
First operation in megawatt rvange Jan. 24, 1966
Reached full power May 23, 1966
Completed 30-day run Jan. 14, 1967
Completed 3-month run Apr. 28, 1967
Completed 6-month run Mar. 20, 1968

End of nuclear operation with 235y Mar. 26, 1968

 
69

The reactivity balance calculations were performed from the start of
reactor operation at significant power. During the very early stages of
the operation, many of the calculations were done manually while the com—
puter program was being checked out. Such calculations were quite feasi-
ble at that time because the terms which depended on integrated power were
negligibly small. Subsequently, the on-line computer was used to execute
modified reactivity balances to provide data for evaluating the xenon-
voisoning term. Once the testing phase described below was complete, the
complete reactivity balance was calculated routinely by the computer every
5 min, and these results were used without further modifications during
normal operation. It was found convenient, however, to manually calculate
the dilution effects that occurred when the fuel loop was drained. Since
shutdown operations generally involved a variety of fuel- and flush-salt
transfers, each shutdown needed to be treated as a special case.

The largest of the dilution effects occurred when the reactor was
drained in July 1965, During the zero-power nuclear experiments in the
period preceding this drain, capsules of enriched fuel had been added to
the loop with the sampler~enricher, and by the end of the zero-power ex-
periments the 235y concentration in the primary loop was about 10% greater
than that in the salt heel remaining in the drain tanks. Thus monitoring
of the reactivity changes in the core actually began with this dilution

in uranium concentration following the July drain.

TLow—power calculations

The first operation of the MSRE after the zero-power experiments and
hence the first opportunity to apply the reactivity balance calculation
occurred in December 1965 and January—February 1966, during a series of
low-power experiments. (The intervening period, JulyDecember 1965, was
spent in completing those parts of the system that were required for power
operation.,) The reactor was operated at powers up to 1 MW, and a total of
36.5 MWhr of fission energy was produced.

Since the computer program for the on-line calculation was not ready
for service during the low-power tests, manual calculations were performed.

However, the analytic expression for control rod poisoning and the various
70

reactivity coefficients that were being incorporated in the computer pro-~
gram were applied. Since very little integrated power was produced, the
xenon, samarium, burnup, and other fission product terms were neglected.

The residual reactivity calculated under ithese low-power conditiouns
was very small (+0.01 + 0.01% Sk/k) and could easily be attributed to un-
certainties in the physical inventory of the system. Hence the reactivity
balance appeared to be in close accord with the system conditions just be~
fore the start of power operation,

In addition to verifying the zero-power reactivity balance, the cal-
culations at 1 MW indicated that the power coefficient of reactivity was
less negative than had been calculated and that the xenon poisoning would
be less than expected (see also the discussion in Sect. 3.2). As a result
of these and later findings, experiments were performed to evaluate these

two terms.

Intermediate calculations

 

Operation of the reactor at powers and for times that produced signifi-
cant fission product terms began in April 1966. This operation soon showed
that the xenon term was inadequately treated and that part of the calcula-
tion was temporarily deleted from subsequent computations. The calculation
results from the other terms in the reactivity balance were then used to

aid in the development of an adequate representation of the xenon poisoning.

In order to use the reactivity balance to evaluate xenon poisoning,
it was necessary to assume that there were no other unaccounted-for reac-
tivity effects, This assumption was valid for the relatively short times
involved in the xenon transients. Since most of the data for the xenon
calculation were developed from the reactivity transients after the reac-
tor power was raised or lowered, small errors due to long-term reactivity
effects were of little consequence.

Figure 17 shows the results of reactivity-balance calculations with-
out xenon for all power operation of the reactor between April and July
1966. The reactor power is shown with each reactivity plot for reference
purposes. The reactivity transients associated with the buildup and re-
moval of xenon due to changes in power are clearly displayed. The appar-

ent steady-state xenon poisoning at maximum power was 0.25 to 0.30% Sk/k.
71

 

CRNL-DWG £6-39C82

  

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Fig. 17. Results of modified reactivity balances in MSRE runs 6 and 7.

The large negative-reactivity transient on June 18-19 was caused by
the development of circulating gas in the fuel loop and its effect on the
xenon poisoning. Operating experience had shown that if the fuel-salt
level in the pump tank were allowed to decrease below a given value, the
amount of gas in circulation would increase; hence the observed reactivity
decrease was at first attributed to a decrease in the average fuel density.
However, the level changes in the pump bowl and overflow tank did not indi-
cate a void fraction of sufficient magnitude (V1.5 vol %) to account for

the reactivity change. The resolution of this apparent paradox was not
/2

obtained until later experiments during operation with 23371, These ex-
periments demonstrated that with helium cover gas and a very low circu-
lating void fraction, a small increase in the void fraction would produce

18,38 This was in contrast

a significant increase in the xenon poisoning.
to the behavior predicted by the model described in Section 3.2, Thus the
major part of the reactivity transient on June 18+-19 was actually due to
variations in the xenon poisoning. The reactivity recovered rapidly when
the normal pump-tank level was restored and the excess helium gas and
xenon were stripped out, Although the mechanism for the change was incor-
rectly identified at the time, the response of the reactivity balance in
this event illustrates the sensitivity of this method for detecting minor
anomalies under otherwise normal circumstances.

The reactivity balances calculated for the period shown in Fig. 17
had not been corrected for all long-term nuclide change effects or for
flush—salt dilution. This is illustrated by the apparent increase in the
residual reactivity at zero power when there was no xenon present. (Note
especially the results on April 11, May 9, June 13, and July 1 and 21-23.)
Correcitions for these factors were subsequently applied to the zero~power

balances to evaluate the long~term effects in the reactivity balance,

Complete calculations

 

The complete reactivity balance calculation, including all known ef-
fects, was first applied to the period of reactor operation which began in
October 1966. Figure 18 shows the power history and residual reactivity
results on a day-to-day basis for the next three runs (the reactivity scale
in Fig. 18 is expanded from that of Fig. 17). During steady-state opera-
tion the results show only minor variations in the residual reactivity.
However, in October and November there was still some indication of dis-
agreement between the calculated and actual xenon poisoning, both in the
absolute magnitude of the term and in the transient behavior. Similar re-
sults were observed during the December 1966—January 1967 operations.
During this period, we attempted to improve our description of the transi-
ent xenon behavior, as was described in Section 3.2.

The larger '"spikes" in residual reactivity can all be accounted for

by abnormal reactor conditions, which were not explicitly treated in the
JORNL—-DWG 56—14{944R

 

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DECEMBER, 1955 : JANUARY, 1967

Fig. 18. Results of complete reactivity balance in MSRE runs 8 and 10.

£/
74

reactivity balance model. For example, the spikes on October 10 are as-
sociated with special experiments during which gas bubbles were circulating
with the salt. Fuel-salt circulation was interrupted for 2 1/2 hr om Oc~
tober 16, and no xenon stripping occurred. When circulation and power
operation were resumed, the actual xenon poison level was higher than that
calculated in the reactivity balance, which assumed continuing circulation
and stripping while the power was low. On October 23, the salt level in
the pump tank was at an abnormally high level for a brief period. The
xenon stripping was much less effective in this condition (because the
pump-bowl spray ring was partially submerged), and the xenon poison level
rapidly built up to a higher wvalue. When a more normal salt level was re-

stored, the xenon poisoning returned to the normal value.

The perturbations in residual reactivity during the November operation
resulted from failure of the calculation to adequately describe the xenon
transients. During this run it was necessary to reduce the power on sev-—
eral occasjons because of conditions imposed by the reactor off-gas sys-
tem. In each case the observed xenon behavior was about the same, indi-
cating a longer time constant for xenon stripping than was calculated in
the model. This disparity in the time constants produced the cyclic be-
havior that was observed.

Considerable difficulty was encountered in the operation of the on-
line computer during the last period of operation shown in Fig. 18. As
a result, substantial gaps were produced in the complete reactivity balance
results, However, the available results were in good agreement with ex-
pected behavior. Again, the spikes on December 23 and 24 and January 12
reflect abnormal reactor operations, which resulted in substantial circu-
lating voids. The smaller variations appeared to reflect slight changes
in circulating void fraction or net xenon-stripping efficiency associated
with variations occurring in the fuel-system overpressure.

The extended periods of full-power operation during runs 11 and 12
provided more stringent tests of the reactivity balance model. Runs 11
and 12 increased the integrated power by 16,200 and 7650 MWhr, respective-
ly, to a total logged output of 40,307 MWhr. In addition to the usual
calculations of power- and time—dependent factors, calculations were re-

quired in each of these runs to compensate for 2357 additions made with
75

the reactor at full powexr. The calculated reactivity behavior was highly
satisfactory, and no anomalous reactor behavior (producing an unexplained
variation in residual reactivity) was indicated at any time. However, dur-
ing these runs we detected some modifications required in the method of
calculation to eliminate some errors that developed, as explained below.
Figures 19 and 20 summarize the results of the on-line calculations
during these rums. These results are reproduced exactly as they were gen~
erated from the computer calculations. The density of the plotted data
points has been reduced from the preceding figures (about 2% of the data
points recorded are shown); however, each plotted point is the result of

an individual calculation, and scatter of the plotted points is still an

ORNL-DW5 67~ 2041R3

 

 

 

 
 
 

 

 

   

 

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APRIL 1987 MAY, 1967

Fig. 19. Residual reactivity during MSRE run 11.
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Fig. 20. Residual reactivity during MSRE run 12.

indication of the precision of the calculations at constant reactor opera-
ting conditions. The points at which changes were made to correct errors
referred to above are indicated by notes.

Except for one negative variation caused by increased entrainment of
circulating gas bubbles immediately after a shutdown in run 11 (Fig, 19),
all the calculated values of residual reactivity were between —0.03 and
+0.10% &k/k. An apparent gradual decline in residual reactivity occurred
during the first several weeks of run 1l. Detailed analysis of the indi-
vidual terms revealed two sources of error, One was a gradual downward
drift in the temperature indicated by two of the four thermocouples used
to calculate the average reactor outlet temperature. These two thermo-
couples were eliminated and replaced by one that had not drifted. The
second error was caused by loss of significance in the calculation of the
14395 concentration. In the program, as described in Chapter 3, only the
change in samarium concéntration was computed, and that change was added

to the last value to obtain the current value. As the 149gn concentration
7

approached 857 of its equilibrium value, the incremental concentration
change computed for the 5-min time step between routine reactivity balances
was outside the five-decimal-digit precision of the computer. As a result,
these increments were lost when the concentration was updated. To avoid
using double~precision arithmetic, the program was modified to update only
the 1498m and the 1°1Sm concentrations every 4 hr while the reactor was at
steady power. Summary calculations made off-line were used to verify the
adequacy of this change.

When these corrections were introduced on March 17, the apparent down-
ward drift in reactivity disappeared. At the same time, minor changes were
made in some of the 135Xe stripping parameters to make the calculated
steadv-state xenon poisoning agree more closely with the observed value.

Other small reactivity variations were observed in run 11; for exam-
ple, from March 29 to April 9. These changes were directly correlated to
changes in the helium overpressure on the fuel loop; a pressure increase
of 1 psi led to a reversible reactivity decrease of slightly less than
0.01% &k/k. The mechanism through which pressure and reactivity were cou-

pled was not yet established, however. The direct reactivity effect of

the change in circulating voids caused by a change in absolute pressure
was at least a factor of 10 smaller than the observed effect of pressure
on reactivity. The time constant of the pressure-reactivity effect was
relatively long, suggesting a possible connection with the xenon poisoning.
Fuel additions were made for the first time in run 11 with the reac-
tor at full power, Nine capsules countaining a total of 761 g of 2357 yere
added between April 18 and 21. The reactivity balance results during this
time showed close agreement between the calculated and observed effects of
the additions. Also, the transient effects of the actual fuel additions
were very mild. TFigure 21 shows an on-line plot of the pesition of the
regulating control rod made during a typilcal fuel addition with the reactor
on servo control., Contrel rod movement to compensate for the additional
uranium in the core started about 30 sec after the fuel capsule reached the
pump bowl, and the entire transient was complete about 2 min later. This
indicated rapid melting of the enriching salt and quick, even dispersion
in the circulating fuel. The weights of the emptied fuel capsules indi-
cated that essentially all their contained 235U was transferred to the fuel

loop.
78

ORNL-DWG 67-10432

39 e e s ————— r

38

 

36 |-

REGULATING ROD POSITION
linches withdrawn)
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0 1 2 3 4 2 6
TIME {min)
0 =1150 hr APRIL 20,1967

Fig. 21. Regulating control rod position during fuel addition.

The reactivity balance results in run 12 (Fig. 20) were essentially
the same as those in the preceding run. Minor variations associated with
pressure -and power changes were again observed.

Another series of fuel additions at full power was made in run 12 be-
tween July 19 and 26, This series consisted of 18 capsules containing
1527 g of 235y, The purpose of this large addition was to provide suffi-
cient excess uranium so that a large amount of integrated power could be
produced without intermediate fuel additions. An experimental evaluation
of the uranium isotopic~change effects associated with power operation was
planned, and substantial burnup was required to make the analyses of iso-
topic composition useful. A secondary result of this large fuel addition
(v0.5% 8k/k) was a drastic change in the control rod configuration. At the
end of the additions the separation between the rips of the shim rods and
that of the regulating rod was 15.5 in., whereas the normal separation had
been 4 to 8 in. The variation in apparent residual reactivity as a func-
tion of control rod configuration was reexamined, and an apparent decrease
of 0.027% 8k/k was observed when the more usual configuration was estab-
lished. This was also consistent with an earlier evaluation of the ac-
curacy of the analytic expression used in the computer to calculate con-
trol vod poisoning as a function of rod configuration. (See discussion of
the control rod worth evaluation in Sect. 3.2.)

On August 3 a computer failure occurred which required recalibration

of the analog-signal amplifiers after service was restored. As a result
/9

of this recalibration, there were small shifts in the values of several of
the variables used in the reactivity balance. Errors in reactor-outlet
temperature and regulating-rod position caused a downward shift of 0.03%
Sk/k in the residual reactivity.

Following the end of run 12, routine checks of absolute rod positions
using the fiducial zero indicators in the rod thimbles revealed an apparent
upward shift of about 0.5 in. in rod 1 (the regulating rod). This shift
represented a reactivity effect of +0.02% 8k/k., The evidence indicated
that this shift in position was caused by slippage of the rod drive chain
on one of the sprockets after a rod scram. The exact time the shift oc-
curred is not known, although it probably would have been detected by an
upward shift in the residual reactivity had it occurred during a run, most
likely at the beginning or end of run 12, when there were routine rod
scrams. Corrections were made for this shift in position by the start of
the next run.

Run 13 had barely begun when an oil leak developed in one of the com-
ponent cooling pumps. Rather than start the planned long run without a
standby pump, the reactor was shut down and the pump repaired. A new run
(No. 14) was started on September 20, 1967, and the reactor was not drained
again until March 26, 1968,

Figure 22 summarizes the reactivity balance data taken during rumns 13
and l4. Prior to run 14, the reactor had operated outside a relatively
narrow range of operating conditions for only brief periods of time. The
normal conditions were 1210°F at the reactor outlet, 5 psig helium over-
pressure at the fuel pump, and a narrow range of fuel salt levels in the
pump bowl. Previous deviations from these conditions had shown that at
least temperature and pressure affected the value of the residual reac-
tivity. However, the effects had not been clearly defined,

During run 14, systematic tests were undertaken to evaluate the ef-
fects of fuel system temperature and overpressure and fuel-pump~bowl level
on residual reactivity. In these tests the reactor outlet temperature was
varied between 1180 and 1225°F and the overpressure between 3 and 9 psig.
Since the fuel-punmp level was restricted by the pump hydraulic performance,
only the normal range of variation was allowed. These tests were performed
at a reactor power of 5 MW to provide the required latitude for the tem-

perature changes.
(% &k/k)

80

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0.05 T e | ol | | L ' 5 | < o ¢ Ve 2 X Ll 2L 1° 505 .
0 %-..o"—"rr‘—f— e REACTIVITY ; @ | o | R VA 2 ot TN
O O = i l. ; l “'lnflt’..i..n'fcgi..‘i.. ....‘.Muln. ' ~._l.‘..‘. ;i‘*... .o i..". i/':?:f:c¢~...:‘..._+..-fis 9 _005 l:.‘o.:.l.‘..."‘ ."...,._:.. .' i }, ; ay 3 -~ —O s 1 O ; E ACT I\'I I - Y -
— 5 Seaes f““'o."".‘ .. : 3 ay (P00 ; QO *“ae . E ° .l...t.n senes '.m' ; 1
| " FUEL PUMP LEVEL '; AR 4 T g | ol | REACTIVITY St 045
-01C 1 |NCREASE DUE TO SALT | P : _ods | . | : | ‘4 _goo [Low PRESSURE '
_015 | RECOVERY FROM OVERFLCW | : | T ~ ool
. TANK 'z | |- . -0.20 ; o e ; 1
“020 -0.25 1T L | | |GH_PRESSURE -
8 ' {9 psig) l
- POWER 8 _
~ © & =
= — =
s 4 = =2
s 2
0 ¢ 12 14 16 18 20 22 24 28 1
2 i ¢
5 17 19 21 23 25 27 29 |1 3 5 7 9 41 13 0 : e L -5 s = 7 o 11
DECEMBER 1967 JANUARY 1968
.05 ' ! [ ‘ [ % L b
OOO el | I fT | 1{ ' 1 a 0.25 0.05
= | | L i |t s i TEMPERATUR
2005 [ “"i"’“fi‘i"it_"—i"'?'"".".l':”'?""'?'-fi...... A e b e 020 e o ©
© i - L REACTIVITY = 015 LOW TEMPERATURE_____ | LOW TEMPERATURE = -0.05
.~C.10 EACTHI (1195°F) "  {1180°F) " N
& i 01C s P Lo i P C | i « -0.10 T
-0.15 l § < 005 % : % ¥ P 7 ¢ F ¥ 2 =045 EACTIVI
-0.20 < 0 -0.20
8 “ RESSURE CONTROLLER | ,i .. | -
2 -005 DISTURBANCE ————& A = s ~0.25
6 _Oqo i ! : gt .t
R POWER | REACTIVITY | . 8
= -0.15 6
2 -0.20 Z 4
0 ., =
t5 47 12 21 23 28 27 29  3i1 3 5 7 9 1 t3 -0.25 \ | IGH PRESSURE ! | - > BOWER
OCTOBER 1967 NOVEMBER 1967 8 (9 psig) LO\EVZ iEtSSL! 0 ‘
6 M2 P {3 15 {7 19 21 23 25 27 29
* s, MARCH 1968
.....I.'.. ._%
2
REACTIVITY o

    

13 15 17 19 21 23 25 27 29 311 3 5 7 2 14

 

JANUARY 1868 FEBRUARY 1968

    

   
     

14 6 18 20 22 24 26 28 301 3 5 7 S i 13
NOVEMBER 1967 DECEMBER 1967
Fig. 22. Residual reactivity during MSRE runs 13 and 14.
81

Although a major part of run 14 was devoted to exploration of the ef-
fect of the system operating parameters on residual reactivity, the run
was started with the normal operating conditions, This operation indi-~
cated that the zenon reactivity behavior, which had been closely duplicated
by the calculation for both transient and steady-state conditions during
run 12, was now different. It was apparent that the steady-state xenon
poisoning was now larger even though there were no observable differences
in the rest of the system. Also the residual reactivity decreased as salt
transferred from the pump bowl to the cverflow tank* and then recovered
after the salt was returned to the pump bowl, producing a slight cyclic
behavior in the residual reactivity history (Fig. 22). This cyclic be~
havior under normal operating conditions was observed on several occasions
during this run (September 23 to October 3, October 27 to November 13,
November 26 to December 15, and February 18 to February 26). It was also
superimposed on effects associated with systematic changes in temperature
and pressure conditions (see Fig. 22). As was described above in connec-
tion with observations during run 7 (June 1966), the evidence now indicates
that this behavior was due to increases in the xenon poisoning associated
with decreases in the salt level in the fuel-pump bowl. As the salt level
was lowered, the amount of undissolved helium in circulation increased; in
turn, this led to slight changes in the distribution and the effectiveness
of removal of the xenon, thereby increasing its poisoning effect. The
early observation of this cyclic behavior provided further impetus for
additional experimental investigations ia this area, which were carried
out during operations with 433U in the fuel salt.

Correlated with the systematic variations in temperature and pressure,
small but significant variations in residual reactivity were observed. The
notes in Fig. 22 indicate the operating parameter changes associated with
the more prominent reactivity variations.

Between the intervals of scheduled off-normal operating conditions
indicated in the figure, normal temperature-pressure conditions were re-
established (1210°F, 5 psig). From the directions of these reactivity vari-

ations, their magnitudes, and the approximate time constants in seeking new

 

See ref., 19 for a detailed discussion of salt transfer to the over-
flow tank and related phenomena.
82

levels, these systematic variations also appeared to be associated with
changes in 13°Xe poisoning. The maximum "shift" in residual reactivity
level observed during any extended time period was 0.15 to 0.20% Sk/k, and
the rates of variation were approximately reproducible (e.g., the tempera-
ture reductions on December 19, 1967, and March 2, 1968, caused the reac-
tivity to decrease by 0.0020 to 0.0025% 8k/k per hour). Since the proces~-
ses of ingesting the undissolved gas and that of gas stripping occurring
in the fuel-pump bowl as well as the behavior of the gas in civculation
could be expected to be sensitive to temperature and pressure, the secon-
dary changes induced in xenon distribution and gas-stripping effectiveness
again appeared to be producing the poisoning variations. The calculation
model For 135xe poisoning in the reactivity balance would not have accounted
for such effects because the model was based on assumptions of a fixed
circulating void fraction and bubble-stripping efficiency (Sect. 3.2).

Although the data in Fig. 22 reflect only the differences between the
nominally calculated poisoning and its actual values, the total apparent
135%¢ reactivity could be easily derived from these special tests by sub-
tracting the calculated !3%°fe effect from the reactivity balance and cor-
recting for known variations in the residual reactivity such as long-term
drift effects (samarium changes, burnup-reclated effects, etc.) and fuel
density~related reactivity effects caused by differences in circulating
void fraction. The remaining changes were assigned to 135%e poisoning.

In general, these tests showed that reducing the fuel temperature and
increasing the system overpressure increased the xenon poisoning. (Further
evidence of the pressure effect is indicated in Fig. 22 when, on January 15,
1968, temporary loss in pressure occurred following a pressure controller
disturbance; this was regained about noon on January 16, and the higher
pressure was tveestablished.) At the highest temperatures, the pressure
effect virtually disappeared. Conversely, the temperature effect was
smaller at the lower pressures. The effect of fuel-pump level was much
less pronounced than the temperature and pressure effects, but decreasing
level always led to higher xenon poisoning for the range of levels investi-
gated., At some of the special temperature-pressure conditions, the re-
sponse (increase) of the residual reactivity to salt recovery from the

overflow tank appeared exaggerated relative to normal operating conditions;
83

however, from about January 30 to February 7, 1968, when the lowest tem-
perature and maximum overpressure were maintained, the residual reactivity
appeared quite stable. The available evidence now indicates that the 135%e
poisoning was at or near its maximum value for this particular set of op-
erating conditions, All the tests conducted during run 14 were a prelude
to more extensive studies made during operation with 233U, aimed at better
understanding of the effects of reactor operating conditions on cover-gas
entrainment and xenon transport and removal. Results of these studies will
be reported elsewhere,!®

A significant variation in the amount of undissolved gas in circula-
tion with the fuel salt was also observed during these tests, These changes
were inferred from reactivity effects at zero power, with mo !3°Xe in the
system, by making use of the density coefficient of reactivity of the fuel
salt. Additional qualitative support for wvariations in void fraction was
obtained during power operation from spectral measurements of the inherent

neutron flux noise3?

and also from temperature observations at the reactor
access nozzle. The reactivity balance gave a loss of 0.032% 8k/k between
the condition with the least circulating voids (1225°F and 3 psig over-
pressure) and that with the most (1180°F and 9 psig), thereby implying a
change of 0.15 to 0.2 vol 7 in the void fraction between these two con-
ditions.

It now appears that the fuel pump and spray-removal apparatus were
normally operating near a threshold where changes in the amount of cover
gas being ingested into circulation could be readily induced by small

19 The total cnanges in circulating gas

changes in operating conditions.
volume were quite small; in density-related effects alone, it could only
account for reactivity changes less than 0.47 6k/k. The rest appears as-
sociated with 135Xe peisoning, in influencing its distribution between 1ig-
uid salt, bubbles, and graphite pores, and its efficiency of removal.

The shutdown at the end of run 14, in March 1968, ended nuclear oper-
ation with the 233U loading., The experience that had been accumulated with
the reactivity balance calculations led to the following conclusions:

1, T1In the normal ranges of reactor operating parameters, the mag-

nitude of residual (unaccounted for) reactivity always remained less than

0.1% 8k/k; its rate of variation over the sampling intervals was small and
84

essentially random (+0.017% &k/k), giving no indication of instabiliiy of
the fuel composition.

2. The largest "abnormal' variations observed in the residual reac-
tivity were less than 0.2% 8k/k and appeared to be associated with changes
in xenon poisoning and the entrained circulating gas in the core at sched-
uled off-normal system operating conditions. Although not aecounted for
by the on~line calculations, these variations were in accord with the qual-
itative behavior expected under these conditions and hence were not re-
garded as evidence of malfunctions. At no time did the magnitude of re-
sidual reactivity approach the adwministrative limit established at the
start of operations (0.5% &k/k, the approximate value of the delayed-
neutron fraction with the fuel circulating).

3. The apparent stability of other conditions in the core, which
could potentially affect the reactivity, allowed the reactivity balance
to be used as an effective diagnostic tool for studying the behavior of

135%c and effects of cover—gas entrainment in the fuel salt,

5.3 Loong-Term Reactivity Trends

 

Although the records obtained from the on-~line calculations adequate-
ly depict the short-term trends in residual reactivity (time scale ranging
from the sampling interval, 5 min, to several days), some further consid-

erations are necessary in interpreting the long~term reactivity trends

(associated with the entire period of operation with 2357y, Primarily,
this is a consequence of the fact that evaluations of long-term effects

in the on-line calculations were based on nuclear data and salt composi-
tion information in use early in the reactor operating history; alsc, a
thorough analysis of the internal consistency of the various component re-
activity evaluations (Sect. 2 of this report) had suggested certain changes,
described and explained below, in several of the coefficients governing the
conversion of concentration changes to associated reactivity effects.

These refinements and revisions in data and calculation rules motivated a
reexamination of the long~term reactivity trends during 235y operations,
after these operations had been terminated. In order to best exhibit the
long—term trends, we have considered only the reactivity balance data taken

at very low power, where any uncertainties associated with the calculation
85

of 135%e poisoning or temperature distribution effects have minimum influ-
ence on the analysis.

The particular modifications in the reactivity balance calculations,
suggested by information and evidence accumulated since the calculation
model was first developed, are summarized as follows:

1. Revisions in nuclear cross-section data combined with self-
consistent calculations of the average reaction rates over all fuel salt
exposed to the neutron flux predicted an increase of about 1% in the burn-
up rate of 235y per megawatt-hour of fission energy and an increase of
about 20% in the net rate of depletion of 238U (with a corresponding in-
crease in the %3%py production rate); a decrease of nearly 11% in the
average capture~to-fission ratio for 23%Pu with respect to the earlier
calculations was also predicted,

2. All terms in the reactivity balance with magnitude based on ear-
lier evaluations of the zero-power rod calibration experiments!! should be
multiplied by a factor of 1.06. This accounts for both an increase in the
calculated delayed-neutron effectiveness and also a small change in the
absolute delay fraction for “43°U to be consistent .with currently accepted

evaluations.?®

The net correction was applied to the rod poisoning, the
excess uranjum reactivity (relative to minimum critical loading), and the

temperature level reactivity (relative to 1200°F).

3. A second correction needed tc be applied to the 235U concentra-
tion coefficient of reactivity, by multiplying this coefficient by 1.072.
This correction made the treatment of the excess uranium reactivity and
the rod~poisoning term self-consistent in the reactivity balance calcula-
tions,

The sources of the cross-section revisions listed under modification 1
were fully described in Ref, 30. The component associated with basic data
and associated neutron spectrum reevaluations was shown by the comparison
of columns 2 and 3 in Table 1, Chapter 4, of this report; the remainder of
the changes are a result of refinements in the methods of calculating the
reactor—-average reaction cross sections. The net revisions affected the
reactivity balance mainly in the calculation of nuclide inventory changes,
with the exception of the revision in the capture~to-fission ratio data for
23%pu, The latter primarily affected the conversion of inventory changes

to the corresponding reactivity addition.
86

The source of modification 2 was a recent reevaluation of the "delayed-
neutron effectiveness factor," described in Section 4.5. As discussed
there, the net result of this examination was a renormalization of all
measured reactivity coefficients in the reactivity balance model.

Modification 3 has its origin in the dependence of the excess 235y
reactivity (and also the control rod worth) on the total 235y concentration.
The needed revision was the result of using, for the earlier calculations,
an average concentration coefficient of reactivity appropriate to the maxi-
mum range of variation in the 2357 loading rather than the more accurate
relation expressed by Eq. (37). This correction was independent of the
additional correction for delayed-neutron effectiveness described above.

The results of including all changes described in the categories
above in the interpretation of the reactivity balance data are shown in
Fig. 23. This evaluation was based on an assumed full-power level of
7.25 MW.;'c In the figure, we have attempted to distinguish the data points
based on measurements taken during separate power rums. Transfer and mix-
ing effects during drain and flush operations between runs introduced vari-
ations in composition of the salt which generally had larger uncertainties
associated with them than did the changes produced during the runs, It is
also likely that the approximate treatment of these effects produced some
of the small positive shift in the residual reactivity observed during the
early power runs (MSRE runs 4 through 6).

The data in Fig. 23 indicate that a slight downward trend in residual
reactivity occurred during the first part of operations with 235U, followed
by an increase during run 14. The statistical spread in the data points

logged over the entire period precludes assigmment of a precise shape to

 

A recent critical review was made of all experimental evidence based
on nuclide changes, both from operations with 2357 and 233U in the fuel
salt, regarding evaluation of the integrated power in the MSRE.*9 The
review recommended that the recoverable fission energy with 235y fuel
should have been 203.2 + 0.5, about 1.7% higher than the value originally
used for calculations., With this fission energy, the nominal full power
indicated by nuclide change measurements was 7.34 + 0,09 MW, This was
sufficiently close to the 7.25-MW value that the uncertainty in power
level was comparable with other uncertainties in the model and thereby
did not modify the conclusions from the analysis.
87

the residual reactivity variation; however, the data taken during the long-
est uninterrupted power run (run 14) definitely indicate that a gradual
increase in the residual reactivity occurred in this interval.

The magnitude and direction of these trends are quite credible in
terms of the basic model, because the calculations of possible reactivity
effects associated with dimensional changes in the graphite indicate that
this type of slow variation could have cccurred (Sect. 3.3). Furthermore,
there is sufficient uncertainty in the degree of removal of low-cross-—
section fission products (Fig. 12) that differences between actual poi-
soning and nominal values assumed for calculations could have produced
gradual reactivity changes of either sign (downward trends could be associ-
ated with less effective removal, and vice versa). For both these reasons,
the magnitudes of residual reactivity variations in Fig. 23 are within a
region in which the basic calculation model is judged to be wvalid,

Those sources of statistical variations and uncertainties in the re-
activity balance data which can be identified include (1) a component of
about *0,01% S8k/k associated with random errors in reading rod positions
and temperatures and with variations between sampling intervals in the

amount of entrained gas circulating with the salt during normal operations,

ORNL-DWG T1-€026

 

0.06

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= - r
= loA MAXIMUM THERMAL POWER, 7.25 MW
@ 0.C4 A vi
& 02 E) !
fi: 0.02 ¢ ¥ . G
§ 0 EI‘ 7 2 o
- |
-0.02 L s -
: : -
x -0.04 g ®
T -0.06
a
a ~—0.08 ]
Lu a)
* -0i10 N SR O T AN T ~ \Q,
O W & » e D
o o \ D 3.0 N \q/ 0% I \0
o Ny \\/\ o "\5&&\ 3 D Q\\\ ¥
o 2 < <« . O % S
a Z IO AV S Qf\ A Q' n,
= DTG DAY O S “ v S
(e ( [
- |
0 20 40 60 80

TIME -INTEGRATED POWER (1000 Mwhr)

Fig. 23. Long—term variations in residual reactivity evaluated at
zero power during 2354 operations; maximum reactor power, 7.25 MW.
88

and (2) a component associated with uncertainties in fuel transfer during
flush operations, increasing in magnitude from zero near the start of power
operation to about *0,02% 8k/k near the end of operation. Any remaining
statistical variations should at present be regarded as inherent in the
technique of recording and analyzing the reactivity balance data, rather
than as evidence of some anomalous physical process in the reactor.

The data summarized in Fig. 23 represent measurements accumulated over
more than two years of power operation of the MSRE. During that time, 235y
equivalent to Vv1.257% 8k/k in reactivity had been depleted in power opera-
tion; 235U equivalent to "0.72% 8k/k had been added to the salt; poisoning
due to 1"%%Sm and '°iSm equivalent to ~0.77% 8k/k had been formed; and
changes in isotopic content of other constituents of the salt had produced
a net reactivity addition of ~0.43% Sk/k. The residual reactivity, evalu-
ated at comnditions of negligible power and xenon poisoning, drifted by less
than 0.1% 8k/k over this period; hence, our accounting procedures indicate

that the reactor behaved in a regular and predictable manner.
89

6. CONCLUSTIONS

The reactivity balance operating history proved quite useful as an
aid in drawing inferences about the core behavior during sustained power
cperation of the reactor. From the conceptual standpoint, the reactivity
balance is a means of viewing the nuclear operations as a process rather
than a sequence of unrelated operating states. As an analysis technique,
it provides an independent monitor of the processes which have significant
influence on the nuclear properties of the core. This may be used in con-
junction with information obtained from other techmniques such as chemical
or mass~-spectrographic analysis of the fuel composition. The disadvantage
of reactivity balance monitoring is that it is highly indirect; only a dif-
ference in the apparent positive and negative compensating reactivity
changes during any operating interval is recorded, and the interpretation
of this difference is a separate problem in operations analysis. An ob-
served residual can be real and due to a single phenomencon in the core or
it can be due to simultaneous small effects, including errors in the basic
models for calculating individual processes. In the former event, other
data given by the reactor instrumentation are important in interpreting
the significance of the residual. In regard to the latter event, we have
given considerable attention in this report to the problem of maintaining
consistency among the component reactivity effects calculated by the model,

The largest uncertainty in the model as developed for the MSRE is
likely to be in calculating the fission product transport and removal from
the core. This model can and undoubtedly will be improved in future ap-
plications of this general technique in nuclear operations analysis for
molten-salt reactors. Despite this uncertainty, however, the reactivity
balance served its intended purposes. As a malfunction detector, reac-
tivity variations less than 0.05% &8k/k were readily observable over short
time intervals. Larger temporary variations that were observed were in
accord with expected behavior of the reactor under operating conditions
then obtaining; at no time did the residual reactivity become sufficient

to have brought operating safety limit considerations into play. As a
90

long~term indicator of the predictability of the reactor nuclear perform-
ance, the residual reactivity remained within the limits of valid pre-
diction of the basic model. In sum, the reactivity balance was a valuable
adjunct for analysis of a new experimental reactor type for which no pre-

ceding experience was available.
10,

11.

1.2 *

13.

14,

15.

16.

R TR T A e e e e e

91

REFERENCES
A. M., Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain
Reactors, p. 600, University of Chicago Press, 1958.

H. P. Daanforth, "The Reactivity Balance as an Auomaly Detector,"
Nucl. Safety, 10(5), 411414 (Sept.~Oct. 1969).

A. F. Henry, "The Application of Reactor Kinetics to the Analysis of
Experiments," Nucl. Sci. Eng. 3(1), 52-70 (January 1958),

E. E. Gross and J., H. Marable, "Static and Dynamic Multiplication
Factors and Their Relation to the Inhour Equation,”" Nucl. Sci. Eng.
7(4), 28191 (April 1960).

T. Gozani, '""The Concept of Reactivity and Its Application to Kinetic
Measurements,' Nukleonik 5(2), 55 (February 1963},

B. Wolfe, "Predicting Reactivity at High Burnup," Nucleonics 16(3),
11621 (March 1958), '

J. Lewins, [MPORTANCE: The Adjoint Function, chap. 5, Pergamon,
New York, 1965,

G. R. Keepin, Physies of Nuclear Kinetics, Addison~Wesley, Reading,
Mass., 1965,

B. E. Prince, Period Measurements on the Molten-Salt Reactor Experi-
ment During Fuel Circulation: Theory and Ezperiment, ORNL-TM-1626
(October 1966).

R. C. Robertson, MSRE Design and Operations Report, Part 1: Description
of Reactor Design, ORNL~TM-728 (January 1965).

B. E. Prince et al., Zero-Power Physics Experimewnts on the Mollten-Salt
Reactor Experiment, ORNL-4233 (February 1968).

MSR Program Semiannu. Frogr. Rep. Feb, 28, 1963, ORNL-4396, p. 3543.
MSR Program Semiannu. Progy. Rep. Feb. 28, 1966, ORNL-3936, pp. 82-87.

B. E. Prince and J. R. Engel, Temperature and Reactivity Coefficient
Averaging in the MSEE, ORNL-TM-379 (October 1962).

R, J. Kedl and A. Houtzeel, Dzvelopment of a Model for Computing 133Xe
Migration in the MSEE, ORNL-4069 (June 1967).

MSR Program Semiarnnu. Progr. Rep. Aug. 31, 1966, ORNL-4037, pp. 13—21.
17.

18.

19.

20.

21.

22,

23.

24,

26.

27.

28,

29,

30.

31.

32.

33.

34,

92

MSR Program Semiannu. Progr. Kep. Feb. 28, 1967, ORNL-411%, pp. 86794,

R. C. Steffy, J. R. Engel, and R. J. Kedl, Xenon Behavior in lthe MSKFE,
to be issued as an ORNL-TM report,

J. R. Engel et al., Spray, Mist, Bubbles, and Foam in the Molten Salt
Reactor Experiment, ORNL-TM-3027 (June 1970).

MSR Program Semiannu. Progr. Rep. Aug. 31, 1967, ORNL-4191, p. 116.
MSR Program Semiannu. Progr. Rep. Aug. 31, 1968, ORNL-4344, p. 36.

H. E. McCoy to C. H. Gabbard, ORNL internal communication, Nov. 6,
1967.

MSR Program Semiannu. Progr. Rep. Aug. 31, 1966, OBRNL~4037, p. 136.

P. N, Haubenreich to H. F. McDuffie, ORNL internal correspondence,
Apr. 10, 1962.

MSE Program Semiannu. Progr. Rep. July 31, 1364, ORNL-3708, p. 376,

J. H. Shaffer to P. N. Haubenreich, ORNL internal correspondemnce,
Jan. 6, 1965,

G. D. Joanou and J, S. Dudek, GAM-I] — A B3z Code for the Calculation
of Fast-Neutron Spectra and Associated Multigroup Constants, GA-4265
(September 1963).

H. C. Honeck, THERMOS — A Thermalization Iransport Theovry Code for
Reactor Lattice Calculations, BNL-5826 (September 1961),

T. B. Fowler et al., EXTERMINATOR-2: A Fortran IV Code for Solving
Multigroup Neutron Diffusion Equations in Two Dimensions, ORNL-4078
(April 1967).

MSE Program Semiannu. FProgr. Rep. Aug. 31, 19689, ORNL-4449, pp. 2226,

L. L. Bennett, Recommended Fission Product Chains for Use In Reactor
Evaluation Studies, ORNL-TM-1658 (September 1966).

T. R. England, Time-Dependent Fission-Product Thermal and Resonance
Adsorption Cross Sections, WAPD-TM~333, Bettis Atomic Power Labora-
tory (November 1962); Addendum No. 1 (January 1965).

P. N. Haubenreich et al., MSRE Design and Operations Report, Fart III:
Nuclear Analysie, ORNL-IM-730, p. 182 (February 1964).

P. N. Haubenreich, Prediction of Effective Yields of Delayed Neutrons
in MSRE, ORNL-TM-380 (October 1962).
35.

369

37.

38.

39.

40.

R T e e T R P N

93

MSE Program Semiannu. Progr. Rep. Aug. 31, 1968, ORNL-4344, pp. 4546.

P. N. Haubenreich and J. R. Engel, "Experience with the Molten-Salt
Reactor Experiment,' Nucl. Appl. Technol. 8(2), 118-36 (February 1970).

C. D. Martin and J. R. Engel, Zxperience with the MSREE On-Line
Computer, to be issued as an ORNL~TM report.

MSR Program Semiannu. Progr. Fep. Aug. 31, 19683, ORNL~4449, p. 10.
D. N. Fry, R. C. Kryter, and J. C. Robinson, Measurement of Helilum
Votd Fraction in the MSEE Fuel Salt Using Neutron-Noilse Analysis,
ORNL~TM-2315 (Aug. 27, 1968).

M5R Progrom Semiannu, Frogr. Rep. Feb. 28, 1970, ORNL~4548, pp. 6566.
95

APPENDIX

Perturbation theory provides a convenient mathematical appavatus for
describing the reactivity effects of arbitrary changes in reactor condi-
tions. We will make use of this general technique, together with the con-
cept of "static" reactivity,* to derive the general relations used in
Chapter 2 of this report. The mathematical treatment will be as brief as
necessary to support the discussion in this report; more complete back-
ground can be obtained by consulting any standard references in the sub-
ject (e.g., Refs. 1 and 7). 1In order to avoid the necessity of referring
to any particular neutronic model, and at the same time simplify some of
the algebraic manipulations, we will make use of a general operator nota~
tion for the neutron balance equation. Thus the basis for the mathematical
description is provided by the equation governing the static reactivity,

O for a particular state of the reactor:

Lo +4¢ = (1 —0o ) P, (A.1)
together with the adjoint equation,
* X *
"y +4y = Q—p) Py, (A.2)

in which ¢ and ¥ represent the direct and adjoint flux fields and [, 4,

and P are linear operators governing the neutron leakage, absorption (in-~
cluding energy transfer by scattering), and production from fission re-
spectively. Once a particular theoretical model is chosen in which to
represent these operators (e.g., multigroup diffusion theory or transport
theory), then the explicit information defining the reactor state, such

as core geometry, material composition, and temperature, completely "fixes"
these operators. The remaining conditions completing the mathematical de-

seription in Eq. (A.1l) arise from the physical conditions for the return

 

* . . .
See discussion in Chap. Z.
96

flux at the boundaries of the reactor and, also, positivity conditions for
the flux field.* Similar conditions, called adjoint boundary conditions,
apply to the solution of Eq. (A.2) for the adjoint flux field. In Eq.
(A.2) the operators with the asterisk represent the adjoint operators,

defined by the inner product relations,

W,Le) = (L v,4) , )
X

(0,40) = (A" v,8) , (A.4)
X

0,20) = (& u,8) , (A.5)

where the symbol (f,g) represents the inner product of the functions F

and g (the multiplication of the functions and integration over the spatial,
energy, and, in certain applications, the directional variables describing
the flux field). The scalar quantity, Pgs is the algebraically largest
eigenvalue of Eq. (A.l) and can be shown to be identical with the largest
eigenvalue of Eq. (A.2).

For what follows, it is useful to notice that the so-called bilinear
identities {Eqs. (A.3), (A.4), and (A.5)] are in the nature of global rela-
tions; that is, they apply to the reactor considered as a whole. In this
sense, they are less restrictive than the equations describing the local
balances, Egs. (A.1l) and (A.2). Thus, for these defining relations to ap-
ply requires only that the direct and adjoint flux functions be chosen from
sets satisfying appropriate boundary conditions, together with certain phys-
ical restrictions such as flux continuity (so that application of the opera-
tors on functions contained in these sets has physical meaning). It follows
that, in the use of relations (A.3) through (A.5), the functions ¢ and ¥
do not have to correspond to the same reactor neutronic state, provided
that the boundary conditions remain fixed and the above restrictions are

satisfied.

 

®
This last condition is necessary to eliminate consideration of
higher—-order eigensolutions of Egqs. (A.1l) and (A.2).
97

In order to derive Eq. (2) of Chapter 2 of this report, we may take
the inner product of the neutronm balance equation [Eq. (A.1)] with a
weight function w, defined on the space of independent variables of the

neutron population,

w,L9) + (,49) = (A =0 )W, , (4.6)
or, by simple algebraic rearrangement,

(w,4¢) + (w,Ld)
p, = 1— . (A.7)
(w,P9)

 

In this case, since the balance Eq. (A.l) holds at each point im the space
of independent variables, the choice of the weight function is arbitrary.
Deliberate choice of w equal to ¢, the solution of the adjoint Eq. (A.2)
for the same reactor state, leads to Eq. (2) of Chapter 2.

Consider now two reactor states, one arbitrarily designated as a ref-
erence state and the other a perturbed, or "“operating" state. Using sub-

scripts a and B to denote these two states, respectively, we may write

L, o, FA 0, = (Lo )P b, (A.8)
L + 4 = (1 — p" A.9
a‘l’a qwa - ( OSOL) LCL ‘Pa . ( . )
LB ¢8 + AB ¢>B = (1 - psB) Py ¢B , (A.10)
SR = 1—o0 )P A.11)
g Vgt Ag b = (o ) Ppdyg o (A

Explicitly representing the perturbation in these operators relative to

the chosen reference state, we may put

L8 = La + 8L, (A.12)
98

I
i

4+ 84,
o

P

It

P+ 5P .
!

(A.13)

(A.14)

Next, we form the inner product of the adjoint flux for the reference state

wa with the operating state Eq. (A.10) and use the defining Eqs. (A.12)

through (A.14) for the perturbations:
(lpD',’ LC!. ¢B) + (lpu$ (Slq)B) + (IPCL’ Aa(bg) + (wu’ 6‘4‘1)6)
= (l fi'DSB) (waa PG ¢B) + (l'm DSB) (was 6P¢B)

By use of the bilinear identities (A.3) and (A.4), we obtain

H

* *
(0o Ly ) + W A 0 (Ly ¥y 0g) + (A, L, 60,

and, from Eq. (A.9),

i

L* A* _ P*
( a was ¢8) + ( o Wu, ¢B) (l de) ( a wus ¢B)

il

1~ p,) (s Pyo)

where the final form of Eq. (A.17) follows from the third bilinear
identity (A.5).
If we now combine Egs. (A.15), (A.16), and (A.17), we obtain

(1= p ) (b Poog) + (b, 6L 60 + (b, 846,
which, after simple algebraic rearrangement, yields

(0o SP05) (b, 0480 (4, Loy
G Poog) Gy Pbg) (W, Bobn)

 

sB m‘psu = - DSB

)

(A.15)

(A.16)

(A.17)

(A.18)

(A.19)
99

This expression is equivalent to Eq. (3a) of Chapter 2.

A slightly different form of the above expression is useful in those
applications where the production operator (i.e., the fissile material con-
centration) is altered. 1In this case, it is convenient to move the oper-

ating state reactivity Py (which is generally an unknown quantity) from

B
the right to the left side of Eq. (A.19). By employing the identity

 

Pag = P T Paq o, in the preceding derivation, one can readily
obtain
SFP SA SL
Pgp ~ Pgq = Lo ) Ei}“’ 2 iB; — Ei“’ > IB; Ewu, 7 iB; (A.20)
“ v e Tt o> Te%) War Ye¥s

equivalent to Eq. (3b) of Chapter 2.

Finally, we may note that Egs. (A.19) and (A.20) are exact, in the
sense that no assumptions have been made concerning the "smallness'" of the
change in the reactor state. Although we are in some applications able to

make what is usually termed the first-order perturbation approximation

 

(i.e., that ¢a = ¢B), it is only in certain instances that the approxima-

tion is justified. The flux fields ¢a and ¢, may in general be quite

different. ’
Tno many cases of interest, such as certain reactivity calibration
measurements, fuel depletions and readditions, and fission product poi-
soning, the perturbations considered have the property that the changes
induced in neutron transport and slowing-down properties of the core are
negligible. In multigroup diffusion theory the leakage perturbation &L
involves only changes in the group diffusion coefficients, and for this
class of perturbations the terms containing 6L may be neglected in Egs.
(A.19) and (A.20). Also for this case, the term with 84 is closely ap~-
proximated by considering only the changes in neutron absorption events.
These observations are quite useful when carrying out approximate theo-
retical calculations of reactivity effects, as described in Section 2.3.
For the class of strong localized perturbations, such as represented
by control rod motions, diffusion theory is inadequate, and more accurate

transport representations are needed for theoretical description of
100

neutron scattering and absorption in the rod regions. In this case, it
can be shown that S§L = 0, but 84 must be understood to include changes in
transfer probabilities in energy and direction for neutron scattering.
Alternatively, recourse can be made to experimental measurements of the

rod reactivity effects, as was described in Section 2.4,
101

ORNL-4674

UC-80 ~— Reactor Technology

Internal Distributicn

 

1. J. L. Anderson 36. H. C, McCurdy

2. R, G. Affel 37. L. E. McNeese

3. H. F. Bauman 38. J. R. McWherter

4, C. F., Baes 39. A. §. Meyer

5. S. E. Beall 40. A. J. Miller

6, L, L. Bennett 41. R. L., Moore

7. E. S. Bettis 42, %, L, Nicholson

8. E. G. Bohlmann 43, L. C. Oakes

9. G. E. Boyd 44, A. M., Perry
10. R. B. Briggs 45. H. B. Piper
11, W. B. Cottrell 46~50. B, E. Prince
12, J. L. Crowley 51. G. L. Ragan
13. F., L. Culler 52-57. M. W. Rosenthal
14, J. R. Distefano 58. Dunlap Scott
15. S. J. Ditto 59. M. R. Sheldon
16, W, P. Eatherly 60. W, H, Sides
17. J. R. Engel 61. M, J. Skinner

18. D, E. Ferguson 62. 0. L. Smith
19. L. M. Ferris 63. I, Spiewak
20, A, P, Fraas 64, D. A, Sundberg

21, C, H, Gabbard 65. R. E. Thoma

22. W, R. Grimes 66, D. B. Trauger

23, A. G, Grindell 67. A. M, Weinberg

24, R, H. Guymon 68. J. R, Weir

25, P, N, Haubenreich 69. M. E. Whatley

26. R. F. Hibbs (¥-12) 70. J. C. White

27. H, W, Hoffman 71. G. D. Whitman

28, P. R, Kasten 72. Gale Young

29, J, J, Keyes 73-75. Central Research Library
30. A, I, Krakoviak 76. Y-12 Document Reference Section
31, M, T. Lundin 77-98. Laboratory Records
32, R, N, Lyon 99, Laboratory Records (RC)
33, H, G. MacPherson

34, R, E. MacPherson

35. H. E. McCoy
100.
101,

102,
103.
104~105,

106.
107,

108,

109.
110.

111.
112-114,
115-116,
117-121,

122,

1.23 .
124~342,

102 o

External Distribution

 

D, F. Cope, AEC-DOSR, 0Oak Ridge, Tennessee

W. H. Hannum, Reactor Physics Branch, Division of Reactor
Development and Technology, USAEC, Washington, D.C. 20545

A. Houtzeel, TNO, 176 Seccnd Ave., Waltham, Massachusetts 02154

Kermit Laughon, AEC-OSR, Oak Ridge, Tennessee

MSBR Program Manager, Division of Reactor Development and
Technology, USAEC, Washington, D.C. 20545

C. L. Matthews, AEC~0SR, Oak Ridge, Tennessee

Yasua Osawa, Atomic Energy Research Laboratory, Hitachi Ltd.,
Ozenji Kawashaki-shi, Kanagawa-ken, Japan

Milton Shaw, Director, Division of Reactor Development and
Technology, USAEC, Washington, D,C., 20545

W. L. Smalley, AEC~CRO, Oak Ridge, Tennessece

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Chattanooga, Tennessee 37401

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USAEC, Washington, D.C. 20545

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20545

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20545

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USAEC, Washington, D.C. 20545

Laboratory and University Division, ORO

Patent Office, AEC-ORO

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